I can take a guess... if Joel had been "in-character" he'd A. Do everything he could to avoid going to Abby's camp which could end up killing the horses and stranding him in a blizzard but hey at least Joel didn't TRUST a person who he saved... B. He could have yelled at Tommy for telling Abby his name but it already happened so... doesn't really change much? C. He could have called Tommy a liar about who he is? Why? I dunno.
As the sequence is presented, they're stuck between a rock and a hard place and the woman they saved offers refuge, Tommy had already told her their names, he was doing so as a calming mechanism in case she was in shock and needed focus, he says their names then asks hers. This is fine for Tommy's character, had Joel done it I could maybe see complaining but Tommy is a nicer, more helpful guy. They had just survived a swarm of bs together and she had saved their lives more than once in addition to them saving theirs, she's by herself at this point and there's no reason to assume their names should trigger anything, like say a need for revenge over something that occurred four years prior. All of this is besides the point of... Tommy says it, not Joel but somehow the argument hinges on JOEL acting out of character.