Mac-sforge motivation for storing pairs
A security definition is stronger if we make it easier for the attacker to win. The point being that the stronger definition rules out even this easier way of winning

In the specific case noted in the question, you are indeed correct.
The scenario you describe is the following:

An attacker queries "Cat" and receives the tag "90". It then computes the tag "138" such that $\mathsf{Vrfy}_k(\text{Cat},138)=1$ and outputs the pair $(\text{Cat},138)$.

The regular $\mathsf{MAC\text{-}forge}$ experiment will reject this output. I.e., the attacker does not win, and therefore this security definition does not rule out the existence of said attacker.

The stronger $\mathsf{MAC\text{-}sforge}$ experiment will accept this output. I.e., the attacker wins, and therefore this security definition rules out the existence of said attacker.