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Of course. But which part the media calls attention to depends on the desired narrative. For example, that Beto O’Rourke is Irish and has not a drop of Hispanic blood is, I think, undisputed. The “Beto” emphasis is designed to rake in Hispanic votes.
I just looked it up. It’s a childhood nickname.
In past I have heard some people criticizing him for using that nickname and I have been curious about it but had never researched it until just now based upon you doing so. As you wrote, it is a childhood nickname. It was given to him as an infant according to Wiki (yeah, I know, not always accurate).
Apparently he has gone by it throughout his entire life. So unless he has planned on running for office since he first soiled a diaper, I really doubt if there is any political element to using it. It is a nickname that stuck and he stuck with it. I know some adults who decided to shrug off childhood nicknames and some who didn't. I think it's just a personal choice.
To me there is no nefarious intent on the part of O'Rourke to keep using the name. Now, do I think it may benefit him with some Hispanic voters? Yes. Absolutely. But that's just luck, not a plot. To my knowledge he has never claimed to have a drop of Hispanic blood. That sort of stupid disingenuous behavior is a trait of another current candidate.
Besides O'Rourke is not the only politician to use a nickname. Nor the only Texan. There is "Ted" Rafael Edward Cruz. Yes, I know that Ted is a somewhat common nickname for Edward but supposedly he first started using that name in junior high school to fit in. Still I would not ascribe any political intent to him doing so. Besides just like O'Rourke, Cruz can't run away from the heritage that his last name portrays. That contrary to their nicknames, they are probably something at least a bit different. And then there was Ronald Reagan, who was called Dutch because his father thought that has a child he looked like a fat little Dutchman. Nothing to do with any Dutch heritage, not that it matters.
So whether someone is using a nickname given to them as a kid for whatever reason, I see nothing wrong with it. So if you're called Jack, Dick, Jerry, Jimmy, Bill, W, Barry, or some other variation of your name or a nickname given for other reasons, there is no reason to have to defend it or change it just because others might want to make political hay.