It’d be interesting if men took on their wife’s first name, so it would immediately indicate to others if a man was married or not. And then there’d be pushback, in the same way there is to the ‘Mrs’ and ‘Miss’ titles, and parents would start naming their sons with female names to start with. (it works less well the other way around, because there’s already girls called ‘Dylan’ and ‘Taylor’, etc).

Here’s one example of this system in use https://en.wikipedia.org/wiki/Icelandic_name

Edit - I realize I probably misunderstood the question. But as said in another comment, it’s just a swap if we’re not talking about patronymic/matronymic naming systems.

In this world I suppose we’d start referring to people by last names more often