Does play a token unit count as a "play a unit"?

Yes, playing a token unit counts as playing a unit.

According to Rule 182.1.a, tokens are played by their owner following all applicable steps for playing a card (or unit), and Rule 350.2 explicitly states that while tokens are not cards, they can still be played. Additionally, RiftJudge FAQ #3881 confirms that playing a token unit counts as playing a unit.