I looked in our attic's subpanel, and there's no ground bar! Because of this, all of the ground wires are connected to the neutral bar. Is it safe?

Do you know by any chance if there is an earthing rod at the property? Some power circuits tie the neutral to the earth and it's a common practise here.

If its a sub-panel, by code the ground & neutral should have separate bars!
But, depending on how old the house is / when the sub-panel was added / etc, it may have been done before that code went into effect, so it'd still be grandfathered in
(what you have will physically work, but there is a chance of creating a 'ground loop' (and thus potential voltage on what should be the ground!))

This will work as long as the supply PEN (the supply Neutral wire in your panel, since it combines the functions of PE - Earth and N - Neutral, is called PEN) is intact, so providing Earth potential to the Neutral bar

If the supply PEN breaks, this fault alone is sufficient to cause shock and fire hazards :

Shock - since current can flow from the Phase, through the appliance loads, to the Neutral bar of the panel, but cannot continue to the supply Neutral. It can, however, continue to the Earth wires and so make all enclosures of all connected appliances live

Fire - same initial conditions as shock, but then, some appliance does have connection to an intact Earth. For example, a luminaire wired from a circuit on that panel and installed on Steel beams in the basement, or a Gas heater with electrical ignition, connected to a Copper Gas pipe going out of the house and into the Earth. Since this connection provides an alternative path to Earth (and therefore to the transformer Neutral), everything will still appear to work normally (voltages on the 2 Phases might be a bit off, but not enough to make anything stop working). You could power up 20A+ worth of appliances (as in - 20A+ difference between the Phases, i.e. this would be the Neutral current), which all goes through a 18AWG Earthing wire of a luminaire in the basement or through a Gas pipe, heating them up...

Therefore, common sense and many electrical codes put requirements and restrictions on the use of combined Earth/Neutrals. They mostly come down to :

 - Requiring to split PEN to PE and N in a single point in the building, most commonly in the main panel, or in a separate connection box upstream of the main panel. This way, the Neutral downstream of this point is only N, so if it breaks somewhere, it won't cause the mentioned dangers

 - Requiring a backup to the Earthing at the point of PEN split. For example, requiring that local Earth rods be connected to the same bar on which PEN is terminated and separate PE and N emerge. This way, if the PEN breaks, it will be backed up by reliable Earth connection

 - Requiring a minimum wire cross section (thick wire) for PEN, so that it isn't a thin wire which is more likely to break, or isn't connected by some small terminal (that have higher chance to not be tightened properly, compared to a big terminal that is tightened with a wrench)

Would it be good to add a grounding bar and attach it to some copper pipe? It would be handy because the air conditioner is right next to the panel.

My understanding in the US is ground and neutral should only be bonded at one place and that place being the panel connected to the utility service or main panel to ensure weird paths to ground don't happen causing things to become live uninetentionally in case of a short to ground.

I actually saw the same thing, at a H-E-B (which is a grocery store in Texas), I saw a completely, DAMAGED exit sign, which a fork lift must've hit it, they're wires coming out of the box, and one of the wires were tagged to ground, which is the neutral that is, on a 277v line.

My understanding in the US is ground and neutral should only be bonded at one place and that place being the panel connected to the utility service or main panel to ensure weird paths to ground don't happen causing things to become live uninetentionally in case of a short to ground.

Yep, thats it!

Would it be good to add a grounding bar and attach it to some copper pipe? It would be handy because the air conditioner is right next to the panel.

No!
What you want to do is add a ground bar, and have it attached to the metal box (to ground the box too)
The neutral bar should then be isolated from the grounded metal box.

Would it be good to add a grounding bar and attach it to some copper pipe? It would be handy because the air conditioner is right next to the panel.

"Some Copper pipe" no

Proper Earthing rods will indeed upgrade safety - from a system that can become dangerous from just 1 broken connection, to a system that won't become dangerous (though it might still be against code, and if you do something to it, it isn't grandfathered in anymore)

Separating Earth and Neutral (providing a separate Earth from the main panel) and ensuring that there is proper backup Earthing there, will provide the best safety (but may be quite complicated to install the additional Earth between the panels)

My understanding in the US is ground and neutral should only be bonded at one place and that place being the panel connected to the utility service or main panel to ensure weird paths to ground don't happen causing things to become live uninetentionally in case of a short to ground.

"Bonding" stands for connecting together metal parts of the building, plumbing, and so on together, to prevent voltages between them. The electrical wiring Earthing must be connected to that too, but the Neutral not (in TT systems there is no connection between Neutral and Earth anywhere in the building at all)