1997 & under accords automatics do shift pretty hard. Remember it is a low powered 4 cylinder thats makes power above 3500rpm and works best @ higher rpm. Does it idle at 900rpm in park or drive/reverse? If the engine is moving a lot its a motor mount/trans mount, could be one of the others if he actually replaced one. A better way to test the mounts besides looking at the engine while engaging into gear is to actually grab the engine & try to move it in all directions. My dads 97 accord 4dr 4 banger will chirp the tires into 2nd sometimes
What color is the transmission fluid & it is full? Check level with engine off parked on a level surface.
Low mile older cars like that usually have lots of dry rotted rubber - hoses, mounts, bushings, etc. because the car wasnt driven enough. My grandpa had a 1990 Cavalier(12 yrs old at the time) with only 12,000 miles, that thing was a pile of shit

I'm not saying your car is junk, but just something may want to check(rubbers).