Key size vs output size of a MAC
The obvious issue is the probability of guessing the key, and the probability of guessing the MAC.

But there's also a qualitative difference.  Suppose you have learned a message/authenticator pair $(m, a)$ where $a = H_k(m)$.


You can test a guess $k_0$ for the key offline by testing whether $a \stackrel?= H_{k_0}(m)$.

The feasibility of finding the key $k$ depends on how much money the adversary has to commit to an offline key search.
You can test a guess $a'_0$ for the authenticator $a' = H_k(m')$ of a forged message $m'$ only by submitting $(m', a'_0)$ online to the legitimate system and seeing how the legitimate system reacts.

The feasibility of forging an authenticator $a'$ this way depends on how much bandwidth the legitimate system has to accept online forgery attempts.