I don't know enough about Azure Stack HCI, though I expect it's similar. For AWS Outposts you can't actually own the hardware. It's hardware leasing on your own premises. More so, it's a black box that you can't inspect, while Oxide's firmware is all open source. You can even flash your own modified firmware without going through Oxide.

iirc azure stack hci is only 1U and 2U configurations, not a full 32U rack.