The Trimble Account management page is only for managing subscription plans where the purchased product is activated by signing in with your Trimble ID. We started offering sign-in plans in 2018 for Shop and in 2019 for Pro. The sign-in subscription plans found in the Trimble Account page do not have or need serial numbers.
Trimble Identity (also called Trimble ID) provides single sign-on access for Trimble and SketchUp online services as well as to authorize SketchUp Pro subscriptions. You can sign-in to one online service and then browse to another online service without a need to sign in again. Starting in 2018 for Shop and 2019 for Pro, some SketchUp products can also be accessed through the Trimble ID.