If you have an animal with one body but 2 heads...

[Mysterian]

...do you give each of the heads a name of their own, or do you call the whole shebang by one overall name?

Asking for a friend.

I personally would give a name per head but am aware of multi-headed mono-named animals such as Cerberus.
YANBU - Name per head.
YABU - Name per shebang.

I'm a man-scientist so would need ponytail and goatee.

Abigail Loraine Hensel and Brittany Lee Hensel (born March 7, 1990) are American conjoined twins. They are dicephalic parapagus twins (having two heads joined to one torso)

One name. Like Fluffy.

With humans it's obviously one per head.

Well, you could also say it's one per body.

How weird that I was thinking this the other day.

1 name.

If it’s a dog or something you call then calling them two names is surely going to confuse them.

They won’t know that their name isn’t Charlie, they’ll hear Charlie and start running and assume that’s their name.

If it’s a dog though you have to call it Fluffy from Harry Potter.

It's two animals that share a body, rather than one animal with two heads - effectively an extreme form of conjoined twin. So I would name each head.

Then there's the obvious questions: Do you call an animal with one head but 2 bodies one name or 2? And then animals with 2 heads and 2 bodies?

Definitely one name per head, ergo, one head 2 bodies = two names

I am gratified & relieved to see a man-scientist eschewing the horrors of the {shudders} MAN BUN. Thank you for your crinigerous service.