History
The reason for the set mix as it exists is that, originally, the dice available were a set of platonic solids, sold by an educational company and repurposed by TSR. Namely, a tetrahedron (d4), cube (square hexahedron, d6), equilateral octohedron (d8), dodecahedron (d12), icosahedron (d20). This was a "platonic solids" set.
D20's were routinely read as d10's in wargaming, and used for generating percentiles, which made use of the d20 as a d10 or pair as a d100 a standard practice, going back as early as 1972.
A 1938 dodecahedron with 2 rounded faces opposite each other was available in 1938 for stock simulations. ➀ It wasn't used by TSR, but shows the probable origins of the pentagonal dipyramid we now think of as the d10.
Names
As to names - the earliest modern set (with dipyramid d10) I can readily find documented by name is the "Dragon Dice" set from 1981 (as evidenced on the packaging copyright notice), as photographed at Dice Collector. ➁
I can, however, cite TSR "catalogue" from a ©1979 TSR product - Swords & Spells lists the older set as "Multi-Sided Dice Set" in the catalogue extract on the back flyleaf. ➂ It reads:
Multi-Sided Dice Sets — Each set contains one 20-, 12-, 8-, 6-, and 4-sided die
The same title and text was used in the 1975 product list in Strategic Review Vol 1. Issue 3 (1975). ➃
So:
Multi-Sided Dice Set - the original d20 d12 d8 d6 d4 platonic set.
Dragon Dice - the "mud dice" set with d10's.
➀ Dice Collector - Mason & Co Stock Exchange Dice - www.dicecollector.com/MINT39_MASON_&_CO_STOCK_EXCHANGE_DICE.jpg
➁ Dice Collector - TSR - www.dicecollector.com/THE_DICE_THEME_TSR.html
➂ Gygax, [E.] Gary, Swords & Spells, 6th printing, Tactical Studies Rules, 1979. Original copyright 1976.
➃ Tactical Studies Rules, Strategic Review, Vol. 1, No. 3, Autumn 1975, Ed. [E.] Gary Gygax. TSR advert on page 8. From the Dragon Archive CD.
Best Answer
Q Workshop manufactures some Japanese dice sets. Amazon has their red and white set for sale.