Reading the manual page http://www.gdal.org/drv_pg.html makes me think that you must add a layer creation option "schema" or dataset open option "active_schema".
Layer creation options
SCHEMA: Set name of schema for new table. Using the same layer name in
different schemas is supported, but not in the public schema and
others. Note that using the -overwrite option of ogr2ogr and -lco
SCHEMA= option at the same time will not work, as the ogr2ogr utility
will not understand that the existing layer must be destroyed in the
specified schema. Use the -nln option of ogr2ogr instead, or better
the active_schema connection string. See below example.
Dataset open options
ACTIVE_SCHEMA=string: Active schema
I would have a try with these commands:
ogr2ogr -f "PostgreSQL" PG:"host=localhost port=5432 dbname=GIS user=postgres password=xxxxxx" "C:\test.gdb -lco schema=my_schema
ogr2ogr -f "PostgreSQL" PG:"host=localhost port=5432 dbname=GIS user=postgres password=xxxxxx" "C:\test.gdb -doo ACTIVE_SCHEMA=my_schema