I have a hypothesis for this behavior, which is that if the client is activated when it's DVR is offline, it does a port scan on the subnet to find a server.
Since I have two DVRs running on the same Windows box, one in a Hyper-V virtual Windows machine (DVR #2), one directly on the host machine (DVR #1), DVR #1 will naturally come up first when host is rebooted. If a client that is normally connected to DVR #2 happens to look for it's DVR while the host is rebooted, it will find DVR #1 instead of #2.
It turns out that I cannot configure multiple subnets nor VLANS with my router, so isolating DVR #2 and it's clients to their own subnet isn't possible (at least not without hacking the router's OS). But, it occurred to me that if I delay the startup of the service for DRV #1 to ensure that DVR #2 comes up first, maybe that would prevent my mom's Fire TV stick from connecting to my DVR. Of course, that may result in some of the other Fire TV sticks connecting to DVR #2 instead of #1, but I can deal with that.
Another possible solution to this dilemma is for the client to recognize it's server by a unique ID, and not connect to a different DVR unless explicitly allowed.
I'll try the delay and let you know what happens. I think I can actually test this hypothesis by rebooting the server and then restarting clients to see what happens...