Well, no, they don't. Android still uses a slightly modified Linux kernel, and there is zero overlap in the drivers. Also they have fundamentally different approaches to task & memory management at the kernel level.
Pedantics aside it's clear that Avatar is not an OS, not even with the most generous and broad definition. At least not with what little they've said so far. Because what they have so far in their diagrams and minimal technical info is yet another JavaScript framework paired with yet another set of web services around user authentication.