APL (Dyalog Unicode), 15 bytes^SBCS

(∧/1=+/)∘.(⊃⍷)⍨ 

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Explanation:

(∧/1=+/)∘.(⊃⍷)⍨ ⍝ Monadic function train   ⍷ ⍝ "Find": Convert the right argument into a boolean vector,   ⍝ where ones correspond to instances of the left argument   ⊃ ⍝ Take the first item of the above vector (i.e., only prefixes)   ∘.( )⍨ ⍝ Commutative outer product: take the above function and apply   ⍝ it for each possible pair of elements in the input   ⍝ If the input is a prefix code, the above should be equivalent  ⍝ to the identity matrix (i.e., each element is a prefix of only itself)   ⍝ To test that it's an identity matrix... ( +/) ⍝ Sum along rows of the matrix  1= ⍝ Test that the sums are equal to 1, results in boolean vector  ∧/  ⍝ And-reduce: test that above vector only contains ones 

APL (Dyalog Unicode), 13 bytes^SBCS

-2 bytes:

≢=∘≢∘⍸∘.(⊃⍷)⍨ 

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Explanation:

≢=∘≢∘⍸∘.(⊃⍷)⍨ ⍝ Monadic function train ⍷ ⍝ "Find": Convert the right argument into a boolean vector, ⍝ where ones correspond to instances of the left argument ⊃ ⍝ Take the first item of the above vector (i.e., only prefixes) ∘.( )⍨ ⍝ Commutative outer product: take the above function and apply ⍝ it for each possible pair of elements in the input ⍝ If the input is a prefix code, the above should have a number of ones  ⍝ equal to the length of the input (i.e., each item is a prefix of only itself) ⍝ To test this... ⍸  ⍝ Find the location of all ones in the above  ≢∘ ⍝ Take the length of the above ≢=∘ ⍝ Compare to the length of the input