We got you, in this case, means we gave you as a gift. It's very common to say that. For example "I got you a little something" or " I got her a new iPhone for Christmas" or "We got you a ticket to a concert for your birthday"

Okay, so it means "... that we bought for you..." It is just lazy casual speech. "<pronoun> got <pronoun>" usually means someone bought or somehow obtained something for someone else. "He went to the ocean and got me a seashell!" I'll also mention that this sentence "That thing with wheels we got you for Christmas?" is a way of implying that 'B' never uses the bike.... which they imply by describing the bike as if 'B' is not familiar with it. This is an example of sarcasm.