The set of all numbers with no repeating digits is sequence A010784 in the Online Encyclopedia of Integer Sequences. On OEIS, such a number is called a xenodrome.
This word is made of the prefix xeno-, meaning "alien," with the root drome, meaning "running, course, race track." It was probably coined by analogy with "palindrome" (with the etymological meaning "running back"), and I suppose the notion is that as you run from one end of the number to the other, every digit you encounter is an alien.
However, this word seems to be extremely rare; it was probably coined by OEIS contributors and it hasn't spread far from there. The only other usage of this word I was able to find in a Google search was the Stack Exchange post "All the Xenodromes," which almost certainly got the word directly from OEIS.