Do our cars really "learn" our driving habits?

Yes, they absolutely do "learn". From the tuning Ive done on my 2012 w/HPTuners, most of the learning that you would notice seems to be on the transmission side of things. But the ECM does "learn" as well - things like knock tendencies, which affects timing, etc. Fuel trims are also "learned", but you wouldn't normally feel that. I'm sure there are other things too, but like I said, the trans learning seems to be what is most noticable to me.

Honestly, my car seems to run better when I reset everything, then it slowly starts shifting worse and worse ("worse" shifting is different to different people, but I tend to find that the trans shifts are smoother and less-noticable after I reset the transmision adaptives). You'd think that it would be the opposite, but at lesat for me, it's not.

I'm curious - did you think the car ran better or worse after the battery change?

It seemed more responsive and quicker to accelerate, I'm an old fart and normally drive pretty easy and try to get the best gas mileage within reason so I wonder if it had sensed that and mellowed the performance a bit due to my easy driving style.
I didn't notice anything different with the transmission but I may have just missed it, I will pay closer attention to the transmission behavior on the way home.
If I drive it harder than normal for the next few days do you think it might stay more peppy than it was before?

I don't think it learns like that. I'm also not sure how long the battery has to be disconnected for it to "forget" everything.

If you use 87 octane gas, I wondering if resetting the knock-learn tables means it's giving you higher timing until it has time to detect and reduce the timing due to repeated knock from the lower octane gas. The 2012+ Impalas really do benefit from higher octance gas - basically, the timing tables start at a high value and get reduced over time if the computer sees repeated knock.

Maybe try running a few tanks of 93 octane and see if the performance stays better - then go back to 87 and see if it deteriorates after a few tanks.