I’ve never seen the clans do the Pete “Maverick” Mitchell style of personal callsign. Remember they already have this whole goddamn thing about earning surnames. Given their focus on proper grammar and speech I imagine they would consider informal nicknames unbecoming.

That said, I don’t really remember the Inner Sphere doing the whole callsign thing either. You will occasionally get nicknames, like “The Black Widow” Natasha Kerensky, or “The Fox” Hanse Davion, but it’s not really a thing in most units. You hear them address each other by given name a lot on the radio. There’s more of a culture of naming 'mechs.

They do seem to have a similar habit to the Inner Sphere of giving stars or binaries unit callsigns. I’m drawing from the Blood Of Kerensky trilogy here, during the Battle of Tukayyid there is this line from Carew (aerospace fighter) to Phelan (Mechwarrior) and Evantha (Elemental):

“Axe One, Hatchet One, this is Quarrel One. We have a target. Keep your head down. We’ll do the work, you pick up the pieces.” The third Point in Carew’s star would be Quarrel Three.

I don’t recall ever hearing any besides unit numbers.