I don't have a car that old but I would almost bet that it originally had some studs on the chrome casting that went in the holes in the dash metal and the entire piece was held by speednuts that captured the studs. The back of the casting looks as if there might have been a broken off stud at each corner and in the middle right in the center of the protrusions at the circled spots. The holes in the dash look too small for a screw but not studs.
Packard used this attachment method on many other trim pieces, emblems and scripts. Many times, if someone is unaware of this configuration they try to pry the piece off and the studs break or pull off the casting in the process.
The other photos show the rear of a script with the studs intact and a selection of typical speednuts. In addition to the flat nuts shown Packard also used round speednuts as well as push in tubular types that could have been inserted in the holes in the dash metal with the studs on the casting pushing into those.
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