Bonnie, you say you have a UK passport and was allowed to vote in the referendum despite not being an Irish citizen (correct?)
Irish citizens are allowed to vote in local, EU, general elections and referenda on the constitution. Non-Irish citizens from the EU are allowed to vote in local and EU elections. That's the situation as far as I know.
People born in Northern Ireland are automatically granted dual citizenship - they of course hold British passports, but may also apply for an Irish one (as far as I ma aware). It may be that you were born in the north, or that the officials at the register assumed you were when they saw your passport, but I'm surprised they didn't check.