WordPress Multisite – is this possible?

It sounds like you have an ideal situation for a multisite. You do not even need to have the sites running the same theme. Your theme however would be exactly the same for each site so you couldn’t modify a theme for an individual one (they are at the base of the multisite not on each site).

The only limits you might find frustrating are that plugins need to be added by the super admin and that some plugins might not work on multisite. Multisites also use more memory and can potentially overtax a database. That said, WordPress.com is essentially one big Multisite so if they can do it …

Here is a decent article on the subject: WordPress Multisite pros and cons