Is it a Cyclic number?

Actually, 18 bytes

;;ru@≈*♂$♂S♂≈╔@S≈=

Try it online! (expects quoted input)

Explanation:

;;ru@≈*♂$♂S♂≈╔@S≈=
;;                  duplicate input twice
  ru                range(1, len(input)+1)
    @≈              convert input to an integer
      *             multiply input by each element in range
       ♂$♂S♂≈       convert each product to a string, sort the digits, and convert back to int
             ╔      uniquify: remove duplicate elements
              @S≈   sort input and convert to int
                 =  compare equality

Python, 86 bytes

lambda n:all(sorted(n)==sorted(str(int(n)*i).zfill(len(n)))for i in range(2,len(n)+1))

Try it online!

Input numbers as strings.

05AB1E, 9 6 bytes

Thanks to Emigna for saving 3 bytes!

ā*€{ïË

Explanation:

ā        # Push range(1, len(input) + 1)
 *       # Multiply by the input
  €{     # Sort each element
    ï    # Convert to int to remove leading zeros
     Ë   # Check if all elements are equal

Uses the 05AB1E encoding. Try it online!