I've noticed that while the European series chokes (Both magnetic and electronic), are rated for very few wattage of fluorescent lamps, often only for a single wattage rating, the American ballasts (Regardless if electronic or magnetic) are often rated for tens, if not less, of lamp wattage rating, which are frequently not compatible to each other in electrical characteristics.
Why this is happens?

European energy standards (since many decades ago - at least the energy crisis in the 70's) do not allow to operate the lamps outside the specified operating conditions (deviate from the standards), so in other words with different than unity ballast factor (so if a 40W lamp is rated to give off 2200lm, it should be operated only on ballasts driving it so it delivers those 2200lm and if it is rated 3000hours, the median life should be at the 300hour on any ballast as well).
The thing is, by operating the lamp at lower current, the light output becomes lower, but the cathode power (big part of the input power) remains the same, so the lamp becomes less efficient. And that is not allowed except dimmed systems operated at reduced power.
At te same time were practically banned all the resistive ballasts (= minimum efficiency became mandatory and the resistive ballasts could not reach that).

On the other hand US practice is to operate the lamps on whatever, just rate the resulting ballast factor figure.

So if you are not bound to so strictly follow the standardized operating figures, you may reduce the production cost by offering a single ballast type for wider range of lamps (instead of a separate model for every lamp type)

Part of the reason I may see in the way broader selection of the different lamp sizes in the US and their greater popularity at homes (= cost sensitive, yet not that much endurance demanding market). Or it is well plausible the narrower selection of the fluorescents in Europe is just the consequence of each type needing a separate ballast.

The other thing I have noted is multi lamp ballast. I don't remember seeing european ballast that can drive more than two lamps but in US there seems to be ballast that can drive at least 4 lamps. Same goes for US two lamp electronic ballasts that can also be used for only one lamp.

The other thing I have noted is multi lamp ballast. I don't remember seeing european ballast that can drive more than two lamps but in US there seems to be ballast that can drive at least 4 lamps. Same goes for US two lamp electronic ballasts that can also be used for only one lamp.

For that the main reason is practicality of the solution, mainly in the historical (non-electronic) context.
In the US for most lamps you need a voltage step up functionality, plus the ballasting, while in Europe on 230V you suffice with just a ballasting impedance (plus a starter).
So for the 120V world it is beneficial to share some of the function blocks among multiple lamps (mainly the step up transforming), as a single transformer is cheaper than multiple smaller ones. So when integrating ballasts for multiple lamps into one box you may save some money by having the components common (mainly the efficiency benefit when integrated with the ballasting inductance into one assembly).
On European 230V each lamp needs its own inductor anyway, so enclosing multiple of them into one box makes no saving, there would be just the same amount of inductors inside. So no benefits against having the inductors separately and be assembled into a common ballast set (with the PFC capacitor, starter sockets,...) by the fixture maker. With separate chokes you then get the advantage of having common parts for very flexible fixture design needs (unusual number of lamps, combining in one fixture each lamp of different type,...).

In modern days (for electronic ballasts) there becomes a requirement that one single failing lamp should not influence the function of more than one other one, while the system should be still robust against those failed lamps. That means you may connect just maximum two lamps per inverter branch (so if one fails and so the inverter shuts down, it shuts down only one good lamp). And because the inverter is the major part of the electronic ballast, the electronic ballasts are designed with maximum two lamp outputs (it does not makes that much sense to share just the rectifier part - it would be very limited cost saving, but quite significant reduction of the use flexibility).
I think the requirement for the lamp influencing is very similar all around the globe.
Some US ballasts treat that in the way the ballast continues to operate even when one lamp fails, but that means high voltage becoming present on the sockets even without the lamps being properly connected, so a safety risk (a lamp inserted by just one end may ignite, so carry the live current onto the other end, where it may be touched by someone) not acceptable in Europe (mainly because people are not used to such fixture behavior - in the US all common F32T8 IS setups behave like that, so people are use to it, so are used to take appropriate care in that respect; by "people" I mean qualified electricians)

And because the inverter is the major part of the electronic ballast, the electronic ballasts are designed with maximum two lamp outputs (it does not makes that much sense to share just the rectifier part - it would be very limited cost saving, but quite significant reduction of the use flexibility)

Most 3x14 / 4x14 T5HE luminaires use one ballast with 4 lamp outputs, and atleast some of them do shut down all 4 lamps if one is bad. From what i seen they are in a 2s+2s configuration with 2 separate output chokes, but the EOL shutdown is main for the entire ballast

@Ash: Roi referred to magnetic ballasts when he referred to multi lamp ballasts, which, of course, non-exists outside the US and Canada.

For that the main reason is practicality of the solution, mainly in the historical (non-electronic) context.
In the US for most lamps you need a voltage step up functionality, plus the ballasting, while in Europe on 230V you suffice with just a ballasting impedance (plus a starter).
So for the 120V world it is beneficial to share some of the function blocks among multiple lamps (mainly the step up transforming), as a single transformer is cheaper than multiple smaller ones. So when integrating ballasts for multiple lamps into one box you may save some money by having the components common (mainly the efficiency benefit when integrated with the ballasting inductance into one assembly).
On European 230V each lamp needs its own inductor anyway, so enclosing multiple of them into one box makes no saving, there would be just the same amount of inductors inside. So no benefits against having the inductors separately and be assembled into a common ballast set (with the PFC capacitor, starter sockets,...) by the fixture maker. With separate chokes you then get the advantage of having common parts for very flexible fixture design needs (unusual number of lamps, combining in one fixture each lamp of different type,...).

In modern days (for electronic ballasts) there becomes a requirement that one single failing lamp should not influence the function of more than one other one, while the system should be still robust against those failed lamps. That means you may connect just maximum two lamps per inverter branch (so if one fails and so the inverter shuts down, it shuts down only one good lamp). And because the inverter is the major part of the electronic ballast, the electronic ballasts are designed with maximum two lamp outputs (it does not makes that much sense to share just the rectifier part - it would be very limited cost saving, but quite significant reduction of the use flexibility).
I think the requirement for the lamp influencing is very similar all around the globe.
Some US ballasts treat that in the way the ballast continues to operate even when one lamp fails, but that means high voltage becoming present on the sockets even without the lamps being properly connected, so a safety risk (a lamp inserted by just one end may ignite, so carry the live current onto the other end, where it may be touched by someone) not acceptable in Europe (mainly because people are not used to such fixture behavior - in the US all common F32T8 IS setups behave like that, so people are use to it, so are used to take appropriate care in that respect; by "people" I mean qualified electricians)

Is it for the same reason why most european fluorescent fixtures have only 2 or 3 lamps per fixture while in US fixtures with 4 or even 6 lamps seems fairly common. I have only seen one 4 lamp fixture type with my own eyes in use and that was from 70's. One supermarket over here has 3 lamp shoplights but other than 4x18w or 4x14wt5 they are max two lamp.

@Ash: Roi referred to magnetic ballasts when he referred to multi lamp ballasts, which, of course, non-exists outside the US and Canada.

More like in generally.

Most 3x14 / 4x14 T5HE luminaires use one ballast with 4 lamp outputs, and atleast some of them do shut down all 4 lamps if one is bad. From what i seen they are in a 2s+2s configuration with 2 separate output chokes, but the EOL shutdown is main for the entire ballast

Actually that regulation came after the moment, when the industry got surprised than one faulty lamp may make the whole x-lamp fixture non working, while many safety standards count on a redundancy when requiring multiple lamps in a fixture. As typical, the legislation heavily lagged behind the tech development...

Is it for the same reason why most european fluorescent fixtures have only 2 or 3 lamps per fixture while in US fixtures with 4 or even 6 lamps seems fairly common. I have only seen one 4 lamp fixture type with my own eyes in use and that was from 70's. One supermarket over here has 3 lamp shoplights but other than 4x18w or 4x14wt5 they are max two lamp.

Well, here the 4xF40T12 were very common till the T12's lost popularity in favor of T8's.
They had 4 chokes, one PFC capacitor and 4 starter sockets...
And 4xF18T8 "ceiling tiles" are very popular till today, along the 3xF18T8 ones.
Both use two chokes and one capacitor, one section is lag ballasted with two lamps in series, the other section is lead ballasted (with the capacitor in series; in te 3-lamper this served the single lamp, in the 4-lamper it serves again two in series).
The 4-lampers became the most common type in the electronic ballast era (two equal 2-lamp ballasts), very frequently replacing the 3-lampers. Say something about energy saving...