This feels like the far more likely option.

The Muse public APIs seem to be heavily inspired by OpenAI's, they even support the richer "responses" API. They have a similarly shaped compaction API, they have the same "encrypted" reasoning. Even the Muse harness seems heavily inspired by (open source) Codex. Considering Azure's historic involvement w/ OpenAI, it seems more plausible that Azure is used as spill-over capacity, and the same Azure infra used to encrypt GPT models encrypts Muse models.

My guess that the generally visible Anthropic/OpenAI details are leftovers from the whole meta "move fast" behavior. There are a few rough edges on the product where it leaks internal codenames (eg. signing in w/ WhatsApp required consenting to using "hatch" on one screen), so I'd believe that this was a leftover on the VM from the prototyping stage, before the Muse models were ready, or because the dev's got to benchmark against different models.