"Bronze" storm door refinishing?

If the bronze finish is an anodized finish you can't readily repair it. Anodizing is an involved process and not something you could do yourself, so you'd have to refinish the entire door by painting. Same case if it's a baked on, powder coated, or even painted finish. You will probably never match the finish. Typical storefront bronze spray paint cans if you want to refinish the whole thing. The door has to be thoroughly cleaned prior to painting to remove wax and oil. It's also a good idea to lightly sand all surfaces to be painted to improve the bond.

Before going to all of that trouble, do some test with a colored Sharpie permanent marker rubbed into some of the scratches. If it's the usual situation, you'll notice the bright aluminum contrasting with the darker bronze much more readily then a black or brown scratch.

The gloss/reflectivity is usually less where the finish has been rubbed/scuffed. Polishing the door with a spray wax will give the door a more uniform gloss and hide a lot of sins.

Give it a shot. You can always paint the door if you don't like the results.