I think it's fair to say that Trinitarianism is polytheistic, but not all Christians are Trinitarians.
Polytheism would be to say that there are three deities with independent minds and wills.
Trinitarianism does not posit this at all. It posits three "
distinct manners of subsisting" or instantiations of the one, self-same divine essence, entity, being and mind in relation to Itself. Not three minds with independent consciousnesses cooperating with one another.
We have to carefully scrutinize what the Fathers (Tertullian, Athanasius etc.) actually meant by
persons here, because it does not carry our modern connotations of independent agency and being.
As I noted above, it means three subsisting and concrete relations of
one divine being, not three 'individual' minds with independent agencies, wills, thoughts and intentions etc. The persons are nothing other than the active relations subsisting in the eternal divine
ousia.
The personality of God, His consciousness, is singular and found in the one
essence (the
ousia) which each Person is, because as St. Thomas Aquinas tells us in the Summa: “
the act of God’s intellect is His substance (essence)” and thus His self-consciousness as an object in Himself is common to the Persons as one 'being', rather than individuated.
I'm not a Modalist, nor am I polytheist but a Trinitarian monotheist. I know what I believe and how to differentiate it from heresies to the left and right of that Nicene orthodox position.
To reiterate:
Trinitarianism - we're talking about three distinct manners of relating of the one and same Being
to Itself, which are called subsisting relations because they actually exist in our theology ('subsist') and are not just different modes in which God manifests Himself to us (modalism). Unity of the one essence is consubstantial between the three
hypostases or relations that subsist/exist in the divine essence, each of which
is that one, singular and self-same supreme reality "God".
Tritheism - three separate divine beings united, like three different human persons, together by a shared plan. Unity of essence can only be meant analogously in this case, not literally.
Modalism - one divine person simply "
manifesting" Himself in different temporary guises in relation to us (His creation) with no real distinction of relations in Himself. This is Unitarianism, God has no distinctions of
hypostases, only the one essence and attributes (i.e. Father, Son and Holy Spirit are merely "masks" He assumes or temporary ways in which he reveals Himself
to us).
Trinity - God relating
to Himself in three distinct subsisting relations,
Tritheism - three gods relating
to one another,
Modalism - God relating
to us in three masks/modes/ways.
To an atheist, I appreciate that this probably seems a bit like "
angels dancing on a pin-head" level stuff but to a theist like myself, the distinctions are extremely important.