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The bumper pans on the front of my '47 coupe use the rubber mounts you show on the right side of your photo. They are fitted vertically as shown in your photo, over the inner steel bumper bracket - one rubber mount on each side of the car's front. I think the mounting slot of my rubber fittings are open on the bottom so they can be slipped down over the top of the steel bracket ..... the bottom having been cut-away/open during a body restoration. Otherwise, one would need to take the steel bracket off to slid the rubber over the end.
My fog lights, usually provided by the dealer as an option, have a tag "Ford" and are stamped manufactured by Unity Company Chicago. They have a fitting that goes thru a hole in the bumper pan and clamps on the outer steel bumper bracket. The fitting that goes thru the bumper pan as a saw-toothed adjustment design to position the horizontal lamp angle.
I don't recognize the fittings you show on the left of your photo; perhaps for an after market fog light ?. They remind me of brackets for radiator mounting.
If you would like photos, I could try that.