Is « Lee » Brennan’s middle or last name ?

So I’m confused about the « structure » of Brennan Lee Mulligan’s multiple names. Is Lee his middle name or part of his last name ?

Googling it, it seems that Lee is his mother’s last name, and Mulligan his father’s. So it would make sense that they are both his last name.

However, I’m remembering (or maybe imagining) someone asking what his middle name is and quickly retracting the question because it was obviously « Lee ». I think it was on game changer (possibly make some noise).

If this memory is correct, do we know if he added Lee as a stage name or if his parents gave it to him ?

Anyways, it’s very much not important, but not knowing has been bothering me for a while. Has he ever said anything about it publicly ?