Using the word hysterical, that old cracker always gets slapped on the table doesn't it, when a woman is anything other than completely cool-girl-relaxed about men doing or saying things that make them uncomfortable. 
Most people within the thread are not saying it is impossible for him to have genuinely picked up her keys by mistake. But are saying more unlikely, given the further string of events and his previous OTT behaviour:
*Flowers, wine & a card (!?!) Before even meeting
*Offering seriously to buy a £500 coffee table for someone you've been on a few dates with.
*Saying 'I don't know what I'd do if this doesn't work out'. Out loud. To the person you've just started dating, that you really don't know well, is batshit. It's creepy and most of all it is manipulative. It is meant to make nice woman think: oh gosh I can't hurt this nice man who has already bought me flowers twice.
Then:
Empties pockets - where we don't know in relation to where OP has put her keys. But unless his keys were also practically directly on top of hers, why didn't he notice there were two or if car keys separate, three diff sets of keys! Again not impossible but then..
Picks up his belongings before leaving, assuming he doesn't use a bag, everything placed in his jeans or coat pocket. Keys carry a weight, is noticeable... Yeah, not impossible that he doesn't notice.
Unsure if he got in a car to go home, if he does, Doesn't notice keys when checking pockets to access key.
Gets all the way home and on entering his own house still doesn't notice he's carrying an extra set.
Once inside takes off coat/clothes and we assume empties pockets and STILL doesn't notice.
OP texts and calls and no answer.
By this point it is at very least questionable.
OP we'd love to know what kind of apology/in depth explanation he gave, besides the keyring similarity, was he acting normally?
Also, quite a few posters have reiterated that this has happened to them, someone they know or someone in the news. I don't know what's hysterical about this outcome also being possible.
To the person who said meet up with him in public and see how she feels...EH? WHY should this be her priority, he has just kept her awake all night long, worrying, losing a day of work, lock changing money. Bad advice for someone who has already said she needs to work on boundaries.
Hope you've managed a nap by now OP!