When I run:
curl -I --http2 https://MYACCOUNT.blob.core.cloudapi.de/MYCONTAINER/MYPATH
I get:
HTTP/1.1 200 OK
...
Does this mean azure blob storage doesn't support http2 at all?
No. Currently, Azure blob does not support http2. You could check this feedback.
But now Azure CDN supports http/2 currently, see this link.