After further inspection I noticed that tables showing in Geodatabase.FeatureTables property is somehow based on the GDB_ServiceItems table in the .geodatabase-file.
If the .mxd-document used in creating the Runtime Content has tables with geometry, these tables are listed on this GDB_ServiceItems table and only them can be used through Geodatabase.FeatureTables property.
However, if the .mxd-document used in creating the runtime content only has attribute tables on it that have relationships to these tables with geometries, GDB_ServiceItems table does not exist inside the .geodatabase-file and all the tables are visible inside the Geodatabase.FeatureTables property. In this case, the tables do not have ServiceInfo and therefore give that "Schema info not found" error.
I can also replicate this by creating Runtime Content with geometry tables that have relationships to attribute tables. After creating the content, I open the Geodatabase and check that only these geometry tables are visible inside FeatureTables property. Then I close my application and open the .geodatabase in SQLite Browser and manually delete all rows inside GDB_ServiceItems. Then I check Geodatabase.FeatureTables and its null. Then I again open the .geodatabase in SQLite Browser and delete the whole GDB_ServiceItems table and after that when I check Geodatabase.FeatureTables, all the geometry and attribute tables are visible there but do not have ServiceInfo.
Should these attribute tables in .mxd-document also generate rows to this GDB_ServiceItems table inside the .geodatabase?