An investor is in several companies' data rooms. Do they sign in to each one?

No. At galley.1936ai.com they sign in once and see Your data rooms: every company on Galley that has invited them. Opening one takes them straight in, and the company's name at the top switches between rooms without another code.

Each room stays separate: its own NDA, documents, access dates and activity. A company never sees which other data rooms an investor is in.