Dyalog APL 16.0, 43 37 bytes

Prompts for comment character, then for code.

↑(↓↑((-(⍸'|'∘=),≢)↑¨'|'∘≠⊆⊢)⍞),¨⊂¯2↑⍞

APL (Dyalog Unicode), 28 bytes

Full program. Prompts for code, then for comment character.

⍞,⍤1⍨⎕FMT⎕TC[2]@('|'∘=)⍞,' '

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⍞,' ' append a space the to prompted-for code

⎕TC[2]@(…) place the 2nd Terminal Control character (Line Feed) at the positions indicated by:

'|'∘= the mask indicating where the pipe symbol is equal to the characters

⎕FMT ForMaT as a character matrix, evaluating terminal control characters (Line Feed moves the "cursor" down one line)

⍞,⍤1⍨ prompt for comment character and append it to each line (lit. to each 1D sub-array)

Dyalog APL 16.0 (non-competing), 43 37 bytes

Prompts for comment character, then for code.

↑(↓↑((-(⍸'|'∘=),≢)↑¨'|'∘≠⊆⊢)⍞),¨⊂¯2↑⍞

Non-competing because version 16.0's is newer than this challenge.

Dyalog APL 16.0, 43 37 bytes

Prompts for comment character, then for code.

↑(↓↑((-(⍸'|'∘=),≢)↑¨'|'∘≠⊆⊢)⍞),¨⊂¯2↑⍞

#Dyalog APL 16.0 (non-competing), 43 37 bytes

Prompts for comment character, then for code.

↑(↓↑((-(⍸'|'∘=),≢)↑¨'|'∘≠⊆⊢)⍞),¨⊂¯2↑⍞

Non-competing because version 16.0's is newer than this challenge.

Dyalog APL 16.0 (non-competing), 43 37 bytes

Prompts for comment character, then for code.

↑(↓↑((-(⍸'|'∘=),≢)↑¨'|'∘≠⊆⊢)⍞),¨⊂¯2↑⍞

Non-competing because version 16.0's is newer than this challenge.