Is there some ingredient in common?
Turpentine is a common ingredient in metal polishes. Aren’t birch trees of the same family as pine (from which turpentine is derived)? That would explain why they have similar scents.
Birch beer gets most of its flavor from natural or synthetic methyl salicylate, also known as sweet birch oil or wintergreen oil. I don’t use silver polish, but I’ll presume methyl salicylate or a chemical with a similar smell is added to create a pleasant odor or to mask an unpleasant one. Pepto-bismol has a similar odor, I suppose because it contains bismuth salicylate as the active ingredient and sodium salicylate as one of the inactive ingredients.
Birch trees are angiosperms (flowering plants) and so are not closely related to pine trees, which are gymnosperms.
Thanks for the info – I don’t have any silver polish in the house at the moment, I’ll try to remember to check the label next time I’m in the store.