Correctly show a selected table (in the data selector) which comes from a multi-schema database (#39930)

* Show selected table in the data selector

Fixes #39807.

* Build schema information when table is selected

* Remove now unused method

* Update only the selected schema id information

* Obtain schema id from the state

* Introduce multiple schemas to the writable database

* Add E2E reproduction for #39807

* Fix unit test

* Fetch schema if needed according to the new condition