Those are BOTH neutral bars.
We can tell, because the panel has what looks like 16 spaces. They must provide 1 neutral for each space in the panel, since you are not allowed (by NEC) to double-tap neutrals on a screw.
When you count the number of non-attachment neutral lugs, I count exactly 16. So there you go.
They didn't give you any ground bars in this panel.
Given that they also didn't give you any room for grounds on the neutral bar, we can discern that this panel pre-dates the requirement for grounds. However, that is fine; this Square D "QO" which is an absolutely first-rate panel there is no reason to judge it for being old.
For now, it will suffice to add the EVSE neutral and ground to the neutral bar - since this is the main panel, (or wired as a main panel), grounds are allowed on the neutral bar here. However, grab the right ground bar - you'll need to go shopping anyway. I bet the model listed on the panel label is still made, and fits the mounting holes already drilled and tapped into your panel chassis (bottom edge of picture, little nubs, probably similar nubs off top of picture as well).
It looks like all your wiring is either AC cable (whose shell is ground), or THHN in conduit (noting writing on some neutral wires). That's why you don't have any grounds now. Check your local jurisdiction to see if you are prohibited from using NM (Romex) cable.
You should fix the MWBCs, though!
Note that you have 12 circuits, but only 9 neutrals. That is because three of the circuits are Multi-wire branch circuits (MWBCs) aka shared-neutral - a red and black hot wire sharing a cable with 1 neutral that they share. These MUST be phased a particular way to avoid overloading the neutral wire! (opposite: 240V between the hots). They must also be "handle-tied" so they must throw together. Both to protect maintainers and signal to the next electrician that it is a MWBC. You can buy approved handle-ties for QO breakers. Or you can simply use a 2-pole (240V) breaker.
Note that two of your MWBCs are on "tandem" aka "Double-stuff" breakers. MWBCs on tandems are bad news, because it's very easy to place them incorrectly so the neutral is overloaded.
You need to identify the red-black pairs in each cable, and place them on 2-pole or handle-tied breakers. Talk to your Square D dealer to see if they have handle-ties for those QOT tandem breakers. Otherwise move the three MWBCs to 2-pole breakers (and move any single circuits on full breakers to the tandems).
If I counted correctly, you should finish with 3 2-pole breakers taking 6 spaces, plus 3 tandems taking 3 spaces, so this won't actually take any space in your panel.
However, you should also check to make sure QOT tandem breakers are even allowed in this panel. That too will be on the panel label. If not, they will need to be converted to full-size breakers, taking up 3 more spaces (leaving 4 left).