Yes. As long as you have enough session quota to go around, you can install the same script on as many sites as you'd like.
With this approach, all of your session data will be intermingled in one FullStory account. You can use search and segments to keep everything straight.
If you'd prefer to automatically silo the data, you'll want additional FullStory accounts, each with its own unique script. Keep in mind, each account would be totally independent (billed separately, have different quotas, settings, etc.) although you would still be able to use the same login to access each account. If you'd like to create a new account, please reach out to Support here.