If someone exclusively worships one god, that person believes there is only one god, or else that there are legitimate reasons to choose exclusivity. Acknowledging the legitimacy of other gods makes that only more acute. Certainly there can be reasons -- in fact, in China, only the emperor could offer sacrifice to any god whatsoever. A nobleman had to have some connection, such as regional, with a god, to offer them.
Regional connections were often required if not so explicitly. Romans objected to foreign gods -- sometimes, others they explicitly invited them to Rome for a better cult in order to cut off the cities they were worshiped in. In Japan, the adoption of the Buddha was explicitly opposed on the grounds that it was offensive to the native gods whom they worshiped.
But if the gods are all local -- well, Hippolytus got himself dead by worshiping Artemis exclusively. Aphrodite was ticked off. To be sure, though the Spartans were noted as very religious in their day, but there is no record of their having any rites in honor of Dionysus or Demeter -- presumably agricultural gods were not the sphere of the Spartans proper, as opposed to the helots who did all the agricultural work. So some spheres might work. Then, sometimes you would send a worshiper off to a more appropriate god. And rejecting a new god would require some argument about spheres.
In particular, a priest sneering at a fad about people for a new cult, bound to fade, would require some argument that this is a false god in some way -- not appropriate for what they want, not regional, or not a god at all. The mere fact that it's a fad would not disqualify it.