RAW
Variant: Familiar on a monster, such as with Imps, Quasits, and Psudeodragons areis an NPC only feature.
The familiar variants in the Monster Manual are for monster and NPC spellcasters. PC spellcasters use the PH.
https://twitter.com/JeremyECrawford/status/725190105451888640
The placement of the new text in Volo's is right before listing the Diviner, Enchanter, Evoker, and Illusionist NPC stats. I think it is clear this variant is meant for those spell casters.
That said, what a DM allows, or not, is up to them at the table.
BalenceBalance
The warlock gets the option of a boon, which can be an improved familiar. Among those options are: Imp, Psuedodragon, Quasit, and Sprite. The innate spell casting isn't too different from the Imp, Quasit, and Sprite's ability to turn invisible, and the Imp and Quasit's ability to shape change... The resistance to all magical damage would worrying, if it had more than 2 HP.
That said, it is an improved familiar that taking costs them more spells or a magic (summonable) weapon. So, to give it to another spell caster without any tradeoff would seem unfair if you have a Warlock at the table.
I think sharing that resisenceresistance would be way overpowered, no matter the class.