Yes - everything submitted to Turnitin will be scanned, stored, and then used as a reference for plagiarism comparison purposes.
One of my pieces of work (interim report for final year project) was marked as being 8% similar, yet the 8% was only the references (as other people would have used the same references as me), my forename (there are plenty of people out there with the same name as me) and the words "Final Year Project - Interim Report". They were all from various different institutions.
So yes - Turnitin doesn't just use the one university that you are studying at, but all the universities and institutions that use Turnitin.