In Ireland (and, I suspect, not only in Ireland) your name is principally a question of fact, not a question of law, and there is no rule that says that you can only have one name. As already pointed out, it’s common for people to use the Irish and English forms of their names in different contexts. At one time, I had one bank account in the Irish version of my name and one in the English version, largely so that the chequebooks would not get confused.
It’s also not uncommon for people to use a name which is neither the name on their birth certificate nor the Irish version of that name. A relative of mine, baptised “Stephen” and so named on his birth certificate, was universally known as “Paddy”, to avoid confusion with other Stephens in the family.
The law takes its cue from this. From a legal point of view, your name is what you are actually known as, and if you are actually known by more than one name, then you have more than one name.
Consequently, Michael can’t change his name simply by saying “my name is Bob now”, He has to get people actually to call him Bob. And, if he has to establish for any legal or official purpose that his name is Bob, what he needs is evidence the people call him Bob.
The more legal and official the context, the more formal evidence you will need. For the most formal purposes (e.g. to get a passport in a name which is not the name on your birth certificate, or the Irish/English form of that name) you may need a to execute and register a Deed Poll, declaring that you were originally named Michael but you are now know as Bob. But, it’s important to realise, the Deed Poll doesn’t legally change your name; it evidences the fact that your name has changed.
Another consequence of this factual approach is that you cannot deny that your name is what people actually call you. If some people still call you Michael, even though you desperately want to be called Bob, then Michael is still your name, or at least one of your names. So, if you are giving evidence in court, and you are asked “Are you Michael So-and-so?”, the answer “no” would be perjury.
The only limit to this is that the law will not recognise or accept name changes effected for the purposes of fraud. I can’t change my name to Bill Gates in the hope of misleading the bank into increasing my credit limit.