IP address via ethernet sometimes in DMZ?

> Model: R7800|Nighthawk X4S AC2600 Wifi Router

   Connected to what?

> [...] I've now twice experienced an issue where my IP address [...]

   "my"?  _You_, personally, have an IP address, or do you mean the IP
address of some computer/device which is connected in some way to
something?

> [...] isn't assigned from my router but my ISP, as though my machine
> is in the DMZ (I have no machines in the DMZ). [...]

   A DMZ server on these routers has a LAN IP address.  Whatever's
happening in your case, it's not like DMZ server on the router.

> [...] The machine is connected to the router via ethernet cable.

   That excludes a few unlikely possibilities.

> [...] Anybody else experienced this, [...]

   It proves little, but I don't remember seeing any similar complaints.

   If you can catch it in the act, then I'd try a little more
exploration of what else it does (or doesn't do), if you can find a way
to talk to it.  My guess would be that the router firmware is getting
bewildered in some way -- flipping into wireless-access-point mode, for
example, or something even less organized.

   Various problems can be caused by bad firmware exhausting some
resource over time.  My D7000 freezes after about a month, for example.
But your symptom is funnier.

   I wouldn't expect a settings reset and manual reconfiguration to
help, but I'd still try it.