05AB1E, 2 (or 3) bytes

C∍

Outputs a mixture of 1s and 0s (depending on the input) for its unary output.

Try it online or verify all test cases.

If a single distinct character for the unary output is mandatory, it's 1 byte longer with multiple options:

• Output as spaces: Cð× - Try it online or verify all test cases;
• Output as 1s: $C× - Try it online or verify all test cases;
• Output as 0s: ÎC× - Try it online or verify all test cases;
• Output as any digit (outputs as a list of characters): CÅ9 (where the 9 could be any other digit) - Try it online or verify all test cases;

C   # Convert the (implicit) input binary-string to a base-10 integer
 ∍  # Extend/shorten the (implicit) binary-string to that length
    # (after which the result is output implicitly)

C   # Convert the (implicit) input binary-string to a base-10 integer
 ð× # Pop and push a string with that many spaces
    # (after which the result is output implicitly)
  
$   # Push 1 and the input binary-stringC×
Î   # Or alternatively, push 0 and the input binary-string
 C  # Convert the binary-string to a base-10 integer
  × # Repeat the 1 or 0 that many times as string
    # (after which the result is output implicitly)

C   # Convert the (implicit) input binary-string to a base-10 integer
 Å9 # Pop and push a list with that many 9s (or any other digit)
    # (after which the result is output implicitly)

05AB1E, 3 bytes

Since outputting the unary as a single character is required, there are a couple of 3 bytes alternatives:

• Output as spaces: Cð× - Try it online or verify all test cases;
• Output as 1s: $C× - Try it online or verify all test cases;
• Output as 0s: ÎC× - Try it online or verify all test cases;
• Output as any digit (as a list of characters): CÅ9 (where the 9 could be any other digit) - Try it online or verify all test cases.

If a mixture of multiple characters for the unary output would have been allowed, since only the length matters in unary, it could have been 2 bytes with:

C∍

Outputs a mixture of 1s and 0s (depending on the input) for its unary output.

Try it online or verify all test cases.

C   # Convert the (implicit) input binary-string to a base-10 integer
 ð× # Pop and push a string with that many spaces
    # (after which the result is output implicitly)
  
$   # Push 1 and the input binary-stringC×
Î   # Or alternatively, push 0 and the input binary-string
 C  # Convert the binary-string to a base-10 integer
  × # Repeat the 1 or 0 that many times as string
    # (after which the result is output implicitly)

C   # Convert the (implicit) input binary-string to a base-10 integer
 Å9 # Pop and push a list with that many 9s (or any other digit)
    # (after which the result is output implicitly)

C   # Convert the (implicit) input binary-string to a base-10 integer
 ∍  # Extend/shorten the (implicit) input binary-string to that length
    # (after which the result is output implicitly)