You're limited to the other Platonic solids [1] for dice having more than the standard 6 faces if you want the odds to be the same, aren't you though?

[1]: https://en.wikipedia.org/wiki/Platonic_solid

In addition to D10, as PeterisP commented, D30 is also doable. Then you can have barrel dice: https://en.wikipedia.org/wiki/Long_dice

Wikipedia generally has quite good overview of non-Platonic dice: https://en.wikipedia.org/wiki/Dice#Rarer_variations

Nope, for example d10 is commonly used, has the same odds for each face, and is not a Platonic solid.

https://en.wikipedia.org/wiki/Pentagonal_trapezohedron#10-si...

Just has to be face-transitive or isohedral for dice: https://en.wikipedia.org/wiki/Isohedral_figure

For example: https://mathsgear.co.uk/products/d120-dice

I should add that I have no affiliation with mathsgear, in fact, I was unaware of their existence until just now. They seem to have lots of nice stuff: https://mathsgear.co.uk/collections/shapes

Since you can assign the same label to multiple faces, it seems like you could construct a fair die that's not a Platonic solid.

I wonder if the rule is that the total face area for each label must be the same?

I don't believe so, the YouTuber standupmaths has an episode on a a three sided coin (die) which is equally likely to land on its edge as either face. They blackboard two theories on what ratio the coin should have, test them both extensively, do proper statistics, and conclude that neither is right and the answer must be somewhere in between.

They are currently encouraging viewers to try a bunch of different values and come up with an experimentally derived result.

I was looking for a 4D dice with 8 sides, just dots from 1 to 4 and couldn't find it (the same one as used in Sorcery's Swindlestones) so I'd definitely be interested.

Not dots, but there are these: https://twitter.com/Meroka_BM/status/840420992048586752

And blank dice are easy enough to get: https://www.amazon.com/Blank-Sided-Educational-Counting-Orga...

I've got the double 4d dice (white) but it didn't occur to me that blank dice and a little bit of paint can just do the job. Thanks!