My surname is "Brennan". Here in the United States, there are Irish-themed gift stores that sell things like claddagh rings, woolen sweaters, and various knick-knacks bearing a design that purports to be the family crest or coat-of-arms for a particular Irish surname. (For example, the "Brennan" one is here: http://www.araltas.com/features/brennan-leinster/). The designs that I've seen are remarkably consistent from store to store, so the stores all seem to be using the same database. But I have no known family connection to Irish royalty or nobility. Is there any legitimate basis for these purported family crests or coat-of-arms?
I blush. I am here editing away my original answer.
I had answered this a few hours ago, but I had answered based on English rules, as I had understood that they were still being followed in Ireland. /u/The_Chieftain_WG points out Irish sept arms in the reply. A Wikipedia article says in a plausible way that there are sept arms that may be displayed in certain ways, but not in others, and by certain persons but not just anyone of the surname. Unfortunately, it being Wikipedia, I don't believe I can even point to it here, and the books I have say little about current Irish practices and don't discuss this point.
"Bucket-shop heraldry" is a plague on heraldry for English arms: made-up arms, or arms stated to be for families (and the English College of Arms is very certain on that point, if you no longer believe me). So on general principles, I'd be very leery of a random site that asserts anything about arms, if they don't provide a citation. In your case in particular, the arms that you point to differ from MacBranain/Brennan arms halfway down this page.
To answer another objection of /u/The_Chieftain_WG: the English College of Arms also grants arms to corporations. The context of the first answer of England's College of Arms FAQ is about arms to a family; non-corporate arms are owned by one person.
My apologies for going past my knowledge, in believing that the rules were largely the same when there is more difference than I knew.