First - Are you in the right place? We started offering sign-in plans in 2018 for Shop and in 2019 for Pro. The subscription plans found in the Trimble Account page do not have or need serial numbers. SketchUp Pro Classic license plans and authorization codes are not shown in the Trimble Account management portal, but rather are found in the License Manager.
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.