It looks to be grouted into space. Normally tiles are caulked when the meet a vertical surface (like a wall or a door frame). The wood should never see grout. Ever.

The outdoor guy tried to fix the big gap that the new frame created (the cut tiles are now quite a bit away from the jam) by using a colour-matched grout. A proper tile setter/flooring expert would not have done that. A colour matched caulk (or a paintable caulk that is painted to match the tile) would be the preference.

This is a "Septic System" guy trying his best to get you a set of doors. Have him come back and grind out the grout in this area. And then see if the freed molding stops the squeaking. And then decide if you want it filled.