I know CRT tube cathodes can sometimes be "rejuvenated", thus restoring some of lost brightness. So I thought can the same be possible to fluorescent lamps cathodes, because both are typically oxide cathodes, but to my knowledge built very differently.

For some reason I have many good and almost unused T12 tubes with cathode filaments in good shape, and they have suddenly stopped emitting, when bad starter was used. The filaments definitely aren't broken, but they still do not emit. Some of these lamps does not even blink in preheat circuits anymore. It is weird. Does anyone have any idea why this happens and can these cathodes be restored some (easy) way?

I also had one philips TLM 20 W tube doing this, and it restored by itself to my surprise, but then it broke when turned on second time (gear was the same all time).

The oxide layer seems similar, but it degrades in very different way.
In the CRT's it get surface contaminated, which could be expelled by higher temperature surge, when a new Ba atomic layer (the thing responsible for the emission itself) on its surface is recreated.
In fluorescents this "reformation" is already happening during normal operation, so the dominant failure mode is when this reformation consumes all the oxide. Therefore once consumed, you can not reform it anymore.

Now if you are talking about some abnormally failed tube, where the emission layer just got contaminated but not consumed, reformation process may bring it back to life. But I'm afraid such reformation is already happening during the failing start attempts in most circuits I know of (the long preheating will provide that heat surge, as well as operating an arc at high cathode drop when the emission is lacking), so if it does not reform, very likely the reason is the emission layer is already spent.
The thing is, long time cathode overheating (stuck starter,...) the cathodes reformation process (oxide -> metallic Ba conversion) went so unecessarily long all the oxide got consumed. Normally this process is stopped by the lamp sucessfuly starting, but with faulty stuck starter it went way too long while the Ba was just evaporating as it got formed.

And normal failure mode is not broken filament, but consumed oxide layer. The filament then breaks only afterwards, when the starter attempting to restart it is overheating it so much it evaporates and physically breaks.

By the way these problems are the reason, why the starter must be replaced with the tube. The old tube damaged it while failing and then that damaged starter would be damaging the new tube, by occassionaly getting stuck in preheat for way longer time than the rated 2 seconds...

Honestly, that was a great question and a great answer. Was a very nice read before bed. Thank you both.