Limit users to specific Segments?

Yes this is possible I think. Ideally each store should have its own Mautic, but I think you can achieve what you want mostly.

Also check out this post

Create your segment emails, in the email set up under advanced set the from address to the correct store. U may run into delivery issues though as the SMTP u are using with Mautic is likely set to use only a single domain. U could work around that by using an email address like - then its sending form the same domain but still unique to sally’s Salon.