Does any girl demand exclusivity? I'm not looking forward to having multiple saves on a game with free-roam sections, but at least I'll know what to expect.
At the moment, not quite. This is probably shifting some. At the moment, the closest to exclusivity is Zelda. To start on her line, you have to go with her instead of Katy (or neither) at a relatively early choice; this doesn't lock you out of all of Katy's future content, but it does keep you from going on 1 date with her and seeing her briefly during your next visit. It also blocks one of your 2 chances to have an extremely brief fling at the Halloween party with one of the guests -- you can get her attention either by tipping very early or by peeking at her during the Katy art date. Note that if you don't go on this date with Zelda, you barely interact with her otherwise; you'll get a sort of awkward exchange at Halloween and possibly a phone call if you make a car purchase later. If she gets more important to the plot shortly (which playing her route indicates she will) I imagine you'll interact with her more, but it's unclear if you'll be at a disadvantage there.
If you pursue her path, you will close yourself out of a romance scene and probably the whole path for Paige later -- Paige explicitly thinks to herself that Zelda doesn't show interest in many guys so she doesn't pursue MC. The walkthrough mod (Scrappy I think?) also indicates that this will block off a future Reina path, although it doesn't stop the date with Reina in the current episode (which is mutually exclusive with another Katy date, or brief non-date conversations with the other girls). Zelda's path also pushes hardest (IMO) against MC's commitment trauma; all of the paths could turn into serious relationships later but Zelda looks a lot like one already, if only because she isn't one for casual connections.
There's some other spots where there is exclusivity for content. Off the top of my head, you have to pick between quality time with Kindra/Artemis and Naomi (which seems like a very important part to a Naomi path, even though that doesn't go far yet), and, as I mentioned, between Katy and Reina for a date scene (or a quick phone call worth relationship points with the others, choose 1). Giving DJ Kink a kiss nets a couple points with her but upsets some of the others (a lot), so it probably works best with a Kindra focused playthrough. There's a very brief scene early in the game where you imagine waking up next to someone, but this does set a flag and is important, at least, for Kindra and Zelda.
At the moment, then, you can basically do one path with Zelda, all the Naomi content, and Reina, and another with all the Katy content, extra time with Kindra/Artemis, and Paige, and get all the major content currently in the game. To absolutely max relationships with any of the girls, though, considering the DJ kiss, 'who do I want to wake up next to," and phone call choices, you'll need to have separate playthroughs for possibly each character so far. Also, as a minor note, there are 3 places where you can get more relationship points with Katy by turning her down for a scene, so bear that in mind for a route with her.