But... it is not. You wouldn't say "mis cumpleaños", but "mi cumpleaños". If I were to ask someone, I'd say "¿Cuando es tu cumpleaños?", so, gramatically, it is singular (Although the plural, now that I think about it, is exactly the same, which is not usual for Spanish... hm).
That said, I am from Spain. Perhaps it is different in South America, or some parts of it? I'd certainly like to know, if that was the case ^^
Come to think of it, we do have both of them as a singular, for example "I found money in my trouser pocket" or "I spilt jam on my trouser leg"
Scissor can be an adjective, for example "scissor door" or it can be a verb, either to do the act of cutting with scissors "I will scissor the paper" or as an act of mutual pleasure between two people, for sexual reasons.
I can explaine it only using 3 languages I know together. In Russian день рожденья is single. But in Ukraine there are two cases of this meaning. Thr First is день народження, same with Russian meaneng, the second is роковини. The second means you have a single day, but in each year. I think that is why 1 single day can have plural form. Because it is repitable.
July 22 used to be a statutory holiday in Poland - Dwudziesty drugi lipca był kiedyś w Polsce wolnym od pracy świętem narodowym
My birthday is July 22 - Mam urodziny dwudziestego drugiego lipca/ Moje urodziny są dwudziestego drugiego (dnia) lipca/ Dwudziestego drugiego lipca są moje urodziny
What was the day yesterday? - Jaki był wczoraj dzień?/ Jaki dzień był wczoraj?
It was Thursday yesterday - Wczoraj był czwartek
What was the date yesterday? - Jaka była wczoraj data?/ Jaka wczoraj była data?
It was June nineteenth yesterday - Wczoraj był dziewiętnasty (dzień) czerwca
Possibly, yes – thing is, in Polish we usually use the verbs(„urodzić”, „narodzić” and their reflexive forms) and verbal nouns(„urodzenie”, „narodzenie”) for 'birth', so it possibly requires some rare contexts to work like that.
„Data urodzin”, when taken literally means 'date of birthday(s)', but obviously aside for year, that's the same date. ;) On official papers where you are supposed to fill in your whole date of birth, the field will be titled „Data urodzenia: [blank space]” or „urodzony dnia: [blank space]”.
Polish encyclopaedias usually when writing date of birth in their entries, use form: „[name], urodzony [DD monthname YYYY] w [placename]”.