Can an irrelevant variable be significant in a regression model? I'm reflecting about statistical significance and I think a variable can be irrelevant and significant at the same time. But, how do I explain that? And if that were the case, should I include that variable to my regression model?

Complementary to Demetri's nice answer (+1):
Aside (unmeasured) confounding, we might have spurious associations that are not the same an unmeasured confounding. Two variables X and Y might be independent but have with similar covariance matrices that looks indistinguishable to confounding due to network or spatial structures. A reasonable fresh reference on the matter is: Network Dependence Can Lead to Spurious Associations and Invalid Inference (2020) by Lee & Ogborn where the authors refer to this phenomenon as "spurious associations due to dependence".
