Especially in the context of Encrypt-then-Authenticate.
I assume that by a "strong MAC" you mean that it is infeasible for an adversary to also find a different MAC tag on a message which has been tagged. Any deterministic MAC (with canonical verification) is a strong MAC, but it is certainly not the case that every MAC is a strong MAC. Thus, indeed, there exist secure MAC schemes (that are existentially unforgeable) but are not strong MACs, meaning that this is a strictly stronger notion (by definition, a strong MAC is existentially unforgeable so the question is just whether it's the same or strictly stronger as a notion). Note that in order to achieve CCA security you need a strong MAC and thus in the context of encrypt-then-authenticate, this is indeed needed. It is possible to define a weaker type of authenticated channel for which it would not be necessary, but typically it is required.
For those not sure what a strong MAC is, see the following pasted excerpt from "Introduction to Modern Cryptography":