First-Name Basis/Comic Books

In Archie Comics' Sonic the Hedgehog, Monkey Khan refers to Sally as Princess until she tells him to call her Sally. Khan responds by insisting on his real name, Ken.
From Blackest Night #5, of all the things in the world:

Bart: Wally! Grandpa!
Barry: Call me Barry or Flash, Bart. "Grandpa" makes me feel old.

And sure enough, not three sentences later do we see that he's dropped it. Considering that Geoff Johns is also writing the new Flash ongoing, it'll likely continue there.

Superman rarely refers to his archnemesis as anything but Luthor. This is also true for everybody who thinks he's a bad guy. Mercy, who loves him, but is employed by him calls him Mr. Luthor.
In ClanDestine, almost all of the Destines address and refer to their father by his first name. The exact reasons are never spelled out, but given that he only seems to have raised few of his children, looks younger than most of them, and that some of them have quarreled with him, it's not that surprising. Not to mention that most of them have established false identities complete with histories that include fictional parents, and might not want to break their cover by calling Adam their dad when someone might hear... The only exceptions are Albert and the twelve-year-old twins.
Squirrel Girl and Wolverine; she's the only one who regularly calls him "James" (rather than "Logan" as everyone else does) and he calls her "Doreen". There are hints that the two used to be very intimate before having a falling out in the past. Of course, any time Squirrel Girl shows up, the reader can expect one or more Noodle Incidents to be mentioned.
Batman's foes the Riddler and the Penguin are often portrayed as being close friends, and refer to each other as "Eddie" and "Ozzie", respectively.

A somewhat common trope in superhero comics as a tool to show intimacy between characters. This is less about being permitted to use someone's real name as knowing what it is to begin with.
In fact, it was a long-running subplot in Young Justice that Robin couldn't give his real name to his friends, because knowing who Robin is might lead to finding out who Batman is.
- When Tim Drake, the third Robin, first started dating Stephanie Brown (AKA Spoiler), he wouldn't give her his real name for the same reasons. When she complained about having to use a codename for her boyfriend, Tim noted that at least Robin was an actual name, instead of something like Green Arrow or Superboy.
Nightwing's most recent civilian girlfriend presented a semi-aversion, which was mildly off-putting for some readers. She insisted on referring to him as "Richard," when he had spent his entire life (that's more or less constant monthly publishing since 1940) going by "Dick." To no one's surprise, she didn't last long.
Actually impossible for Apollo and Midnighter, because they don't have real names - we see their identities being erased during a Flash Back in their first appearance.
Among other reasons, Damian Wayne is disliked because he refers to beloved family butler Alfred as "Pennyworth."