They do when trademark law gets involved, and one company literally changes the name of their character to not be what it used to be to compensate.
It's not like it matters, though. DC reboots its entire continuity so often nearly every single facet of "Shazam"'s personality and backstory is different from what it used to be anyway, so changing his name is about the least significant thing they've completely altered.
I have little sympathy for DC in this situation, though. The only reason they own the former Captain Marvel/current Shazam as a character in the first place is because they sued the company that actually created him out of business and then bought up their assets out of bankruptcy, and then they ignored the character for years because they didn't really give a shit about him, which is why they let trademark lapse. And then, after it became an issue, they still spent years whining about how unfair it was and throwing tantrums rather than just solving the problem the way they ultimately did anyway, by just changing the character's name (to something most people associate with the character more than his actual former name anyway).
How did they sue the company that created the character to bankruptcy? ---
There are more than two genders, try to change my mind