This is an important thing to note

Dating the girl your friend likes

They're either hearing this directly from him or they're picking it up based on his behavior. You've noticed that when you do go places with him he introduces you like he's proud of you, which when you think about it is an interesting move.

But your mutual friends or other people who see you guys interacting might see it plain as day since they don't have any reason not to. You might introduce her to someone with a similar interest or career so that she can network but you're not feeling proud of her accomplishments like they have something to do with you. She knows and she doesn't like it. He might not want to accidentally say anything that he shouldn't be saying.

You might introduceBut your mutual

If a guy was checking you out while you were with him he might even get the vibe that you two were a couple and not bother talking to you. Like you met on Tinder and immediately realized you were meant to be besties. He looks you in the eye when you're talking because he's really listening to you, and he looks happy listening to you because he is. Or he might like the feeling of being with you so much that it's kind of hard on him to go home at the end of the day and deal with the fact that you're not actually his girlfriend.

If you're in a relationship or you don't have feelings for him you might try to ignore the possibility completely. This goes way beyond friend behavior, this is boyfriend behavior.