How come this is plural but translates to a singular? Is it because the literal translate of urodziny is something other than birthday?
while you are right keep in mind that the birthday you celebrate this year is still "urodziny" (It's like Christmas)
Urodziny is one of the Polish words that are always plural. Just like 'drzwi' (a door/doors) or 'spodnie' (trousers)
"Kiedy są ich urodziny?", "Kiedy mają urodziny?" - as "urodziny" are pluralia tantum.