Me and my opponent play Seat of Power, I conquer with my Lonely Poro and use Shuriken Flip to move it to the other battlefield, do I draw when I conquer?

No, you don't draw.

Seat of Power's "When you conquer here" trigger goes on the Chain when you conquer. You can absolutely play Shuriken Flip before it resolves and move the Poro to the other battlefield — but moving there only applies Contested to that battlefield. You don't Control it until the resulting Showdown resolves with your units remaining (Rules 190.4 / 466.5), and that resolution happens after your draw trigger has already come off the Chain.

So when the ability resolves, "each other battlefield you or allies control" = 0, and you draw nothing. For the draw to work, you'd need to actually control another battlefield at that moment — winning the Showdown at the other Seat of Power first would conquer it, but that would trigger that battlefield's own draw ability (and you'd have to keep units on the first one or you'd lose control of it in the next Cleanup).