I'm in the process of setting up monthly memberships myself and getting stuck by this same conundrum.
As far as a I can see right now, yes, you would have to manually keep track of their sessions. There isn't a way, right now, that this can be supported in Square. Bummer? Yes. However, I've been instructed by other Sellers to look into third-party integrations for this, which I haven't done yet because...well...time...
One workaround that I'm finding is to rely on system processes and automation.
For example, depending on what type of Square 'involvement' you have, you can implement an automation trigger train that will send a message to a client alerting them it's time to schedule their appointment, at which point you could send a form that they can choose which service they'd like that week and then book it for them. It's messy with a few extra tasks, but something like this might be the 'best' option if you're wanting to keep it all in the Square ecosystem.
My secondary suggestion would be to submit a feature request. As I saw another Seller say: we can enact the change! Squeaky wheel gets the grease, right? 🙂
We're all trying to do our best to achieve our best. We can help elevate Square and its capabilities with kind vocal suggestions.
If anything else comes up or if I find another workaround I'll do my best to keep you informed.
