You can currently change object subtypes via Conduit (maniphest.edit) but not via the web UI.
Changing object subtypes is inherently a somewhat-perilous operation that likely has a lot of rough edges we'll need to smooth over eventually, mostly around changing an object from subtype X to subtype Y, where some field exists on one but not the other. This isn't a huge issue, just not entirely intuitive.
It should also, in theory, be fairly rare.
As a reasonable middle ground, provide web UI access via the bulk editor. This makes it possible, but doesn't clutter the UI up with a rarely-used option with rough edges.