I like this question! I think it's clear that Asterion has feelings for MC, but I find it difficult to answer the question "could an immortal being form a meaningful relationship with a human who will inevitably pass away?" in general. As you mentioned, it might be difficult to see someone on the same level as you intellectually if you've lived hundreds as many times longer than them. I also wonder how much it would wear on someone to see their loved ones grow old and pass away over and over. Would that make it more painful to form connections, and would they wind up avoiding such connections to avoid the hurt?
I don't see either of these being a problem for Asterion, however. He has plenty to learn about modern society from MC, and plenty to share from his own experiences, which lets them see each other as equals. He has also had few former masters that have treated him well, so he hasn't become hardened by losing generations upon generations of loved ones. He clearly wants to get to know MC and isn't apathetic to the attention given to him.
As a last note, I just finished rereading Circe by Madeline Miller, and that book grapples heavily with the relationships between gods and humans; some gods are clearly hardened by their divinity and don't spare a second thought to humans, and some are intrigued by the imperfections that humans have. I really enjoyed it, and you might, too! (Not to mention that the legend of the Minotaur of Crete is a big event in the book :) I got a kick out of rereading that part after having played Minotaur Hotel!)