Does this -szer ending have any additional meaning besides being used for multiplication?
As a suffix, probably not. It is used for multiplication and, also, when talking about times, like once, twice, three times, etc.
But "szer", as a noun, means a lot of things. I wouldn't know the exact translation of it. It is close to "thing" or "ware" but doesn't exactly mean that. And it loves to be the last part of compound words. But it is not a suffix there. A few examples:
Ékszer - jewelry
Vegyszer - chemicals
Tápszer - feed (noun)
Tanszer - things needed for school
Szertár - storage for stuff, like for things used in physics class