It seems there are two sets of questions interleaved within this discussion.
There are three options for using SQL-Server Express:
- As a Workgroup or Desktop geodatabase ("Database Server" access)
- As an Enterprise geodabase ("Database Connection" access)
- As a simple SQL connection via Query Layers
Any way you roll it, Microsoft licensing restricts the SQL-Server application to a single CPU, a RAM limit, and a total storage limit. Using the non-Enterprise geodatabase options, Esri licensing also applies a 3 or 10 connection limit.
To answer the original question, yes, Microsoft SQL-Server Express is a supported RDBMS platform for an enterprise geodatabase (provided you conform to the system requirements).
It's also possible to use Esri tools to create new Express instances or geodatabase-enable databases within them, but only the enterprise geodatabase tools can create an enterprise geodatabase, and the Workgroup/Desktop tools create non-enterprise dtabases.
- V