there's something interesting about emmrich's hangout being so... structured and composed. everyone else's hangout is under the guise of some sort of errand, yes, but they're very casual and free-form. whereas with emmrich, he's taking you on this tour of the memorial gardens in the necropolis. and it is a tour—when he shows you the undead tableaux, it feels like he's reading off a script with the way he describes it to rook, tour guide style. and later, when he opens up to rook, it is always him initiating with the distinct feeling of something being said on purpose, like he had prepared to discuss it.
the flirt at the end gives a romance-flavoured explanation: that he was specifically trying to impress rook, that this is an attempt to endear himself to them. and in a lot of ways, it does look like that: he takes rook to one of the most beautiful, peaceful places he knows, opens up to them about his past, and then he has tea set up at the end of it, like a good host. (he is also rather caught off guard when you flirt with him here, and it could be interpreted as him being embarrassed at being caught out, or trying to walk it back so he doesn't offend).
however.... i don't really think that was his intention? this is probably coloured by the fact that i feel like, comparatively, emmrich has had very few one on one interactions with rook to this point and he breezes by the only flirt before this, but i don't think he walked into the memorial gardens with any intention to charm or otherwise woo rook, here. i think his intention was to get them alone and get an honest read on how they felt about him. specifically, about his work as a necromancer.
if you've rotated different people through the party with him, and if you've paid attention to the conversations he's had with others in the lighthouse, it becomes abundantly clear basically immediately that everyone has opinions on necromancy, and they largely fall into the territory of distrusting, put off, or even outright disgusted by it. a lot of emmrich's early banter with several party members is him going about the pretty exhausting ordeal of defending his magic, the mourn watch, his fields of study, manfred's existence, etc. and because of the lack of one on one time with rook, i'd be surprised if he felt like he'd really had an opportunity to get a read on what his new colleague actually thinks about it all.
so i think he is trying to impress rook, just... not like that. i think he's trying to prove that the necropolis is beautiful, that it's laden in history and rituals that mean something. that the mourn watch is good (the sort of order that would take in an orphan, and give him purpose and structure), and that even necromancers can fear death. i think he's trying to prove that he's a person, under all the misconceptions about his work that seem to go around the lighthouse when he isn't looking.
and i think that's why he looks so taken aback and genuinely surprised when rook flirts with him at the end—because it really wasn't his intention, but he recovers fast enough to pivot with grace. i don't think he's uninterested, i just think he wasn't sure rook could be.