I don't understand why you'd say that Nana is "warmer" or that Grandma is "formal". The names are all fine, there is no need to compare them.
I think partly it was that no one at school called their grandma, grandma. They had a nan or whatever. And it seemed like they were more affectionate/closer and more informal. But just my perception.
It'd be like calling your mum 'mother' as a child. Well, not that bad. 
"Grandmother' or "Grandmama" would certainly seem weirdly Victorian and formal but not grandma.
True.
I think within reason the grandparents in question should be allowed to choose what they are known as, so long as they don't choose something unbearably twee.
Aw I think having a twee name for your Grandma might be nice. How about 'possum,' 'moomin' or something like that? Or something cottagecore IDK.
I don't recall ever challenging the status quo
Why would you "challenge" the name she liked to be called?
@starrynight21 I suppose not. It just is something that seems/seemed weird to me.
"She wasn't well off financially or anything "
Same with this odd comment - names for grandparents have no connection to class or finances. You're massively overthinking the whole thing. There are dozens of terms to denote grandmotherly status - none of them have any bad connotations, they are just nice terms to show a relationship.
I heard it mentioned the other day as a thing that was supposedly more posh etc. So yes, it is sometimes referred to as if it were a class thing.
You didn't know her even as well as I did.
They were quite poor and she was the kind of person that was concerned a little (though not massively excessively) with things appearing common vs. not common so I assume that was part of it. Like with the nan is a goat comment. So she was the one who thought that had a bad connotation of commonness or disrespectfulness perhaps.
I'm not overthinking it, I was just amused when I heard the sentiment again and wondered if many others had heard it/known someone who was like it.
If I ever said to kids at school that my nanny was visiting, they assumed I had a Mary Poppins childminder
@Goatinthegarden Yes it does sound a bit like that. Children are cruel and most want to fit in. x