We call them "hyphenated", or more poetically, "double-barrelled" surnames in Britain. It was usually done by the aristocracy to preserve both family names after an auspicious, possibly engineered, union.

(The "double-barrelled" terminology derives from, well, double surnames, but also the fact that the aristocracy are the ones who go game hunting with double-barrelled shotguns. There's a "shotgun wedding" pun in there as well, though usually, as I implied, these things are usually by choice (thought not necessarily that of those being married) rather than a hasty wedding.)

More interesting things happen when one or both people getting married already have double-barrelled surnames.

It's how the British royal family ended up with the family name Saxe-Coburg-Gotha, ...at least until they changed it to Windsor because being conspicuously German was not a good look in Britain during the early 20th century.

Anyway, as that example suggests, the usual rule is to pick the order that sounds best and put insert hyphens between the surnames. The Weinersmiths haven't quite followed the same rules, or else they'd spell it "Weiner-Smith", but it still sounds better than any variant of Smith-Weiner (in my opinion).