It means being too demanding or strict in complying with a rule. I guess that you never heard it before because the catholic tradition in your country is much less than in Romania and Spain. Although, at least nowadays, it is used with a little humorous tone 
Quite funny, in France we have almost the same expression.
"We do not need to be more royalist than the King himself...", with the exact same signification. Funny how some expressions were adapted after some special circumstances
Yes, we share a very similar history when it comes to political power in several countries in Europe, and it seems that the same expressions crossed borders many centuries ago without much trouble. To be exact, in Spain it is "to be more papist than the Pope", halfway between royalists and Catholics, I guess
On another note, I remember AoBTers that we have two new translations in progress. No need to rush, just a friendly reminder because maybe it went unnoticed:
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Translations IN PROGRESS: