Does Express store any statistics to indicate when performance is held up due to being Express? I am looking for business reasons to upgrade from Express to Standard. as per my question here

Does SQL Server store internally any information about when it is held up due to edition / licensing features such as number of available cores?

What queries could I run to find this out?
SQL Server isn't going to tell you specifically that there is a performance limitation due to it being Express Edition.  But in general, if you start collecting and monitoring wait statistics, you'll see whether there are any bottlenecks due to insufficient CPU or memory.  So if you've given SQL Server Express the max number of CPUs and memory, then bottlenecks in these areas generally indicate that it's being limited by the limitations of the edition.  Technically, you could give it faster processors to help overcome the CPU limiation, but I assume that's not an option.

See SQL Server Wait Statistics: Tell me where it hurts to get started understanding what you need to be monitoring.