Vyxalr, 6 bytes

ƛƛ⁰O;g

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Explained

ƛƛ⁰O;gƛ       # Map the first inputƛ  ;   # Map for each letter in the word⁰O    # Count how many times the letter appears in the second input     g  # Get the minimum

Vyxalr, 5 bytes

taking words as list of list of characters instead of list of strings

ƛ⁰vOg

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