The classification of religions by how many gods they hold is the oldest comparative scheme in the field and among the least reliable. The categories are useful for description and misleading as a taxonomy, for three reasons.
First, the number of gods is frequently not what a tradition itself regards as its central claim, so classifying by it sorts traditions by something they may consider incidental. Second, the terms were coined for the most part between the seventeenth and nineteenth centuries by European scholars with polemical or apologetic purposes, and they carry those purposes with them. Third, traditions routinely combine what the scheme separates. Hinduism is the standing counter-example, containing at once an unqualified absolute, personal deities addressed as supreme, local powers, and philosophical schools ranging from robust theism to explicit atheism, all under a single heading, and without its practitioners experiencing any difficulty.
The terms also carry evaluative freight from their origins. Nineteenth-century comparative religion typically arranged them as a developmental sequence running from animism through polytheism to monotheism, and treated monotheism as the endpoint of religious evolution and the measure of the rest. That framework has been abandoned, but its vocabulary remains in use, and words such as "animism" and "fetishism" retain traces of it, which is why several of them are now used with care or replaced.