First - Are you in the right place? We started offering sign-in plans in 2018 for Go 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 provides access to all Trimble applications and online services, including all applications which are part of SketchUp subscriptions. With your Trimble Identity, you can sign in to one of your Trimble online services and then browse to another without the need to sign in again.