The landlord only has one ethernet cable coming from the ISP. If I connect the ISP cable, homeowner's router and the new renter's router to an unmanaged switch, then
This depends on how the ISP hardware is setup. If it gives a single public IP to whatever is plugged in, this will not work. If the ISP provided equipment does NAT (ie. gives private addresses), sure you can plug in two NAT routers and they will work just fine with DOUBLE NAT. Some people think it is evil, but it would work fine. Even if the ISP line is more like a cable modem and provides a single public IP, you could put your own NAT router in front of the two "subscriber" routers.
A new renter moved in and wants his own private network with no communication between his and my private network.
In the case above with double NAT, actually you could each spy on each other, because the two "subscriber" routers share a broadcast domain on the WAN port. Anyone who has access to that cable can spy, period. Switches provide cosmetic protection, but it is easily circumvented. There are entire classes of MITM attack that arise from sharing a broadcast domain there. Honestly, it is a little secure, but you need a bit of trust.
The correct "business flavored" way would be to buy a router that supports VLAN ports like a Ubiquiti or Cisco Meraki thing, give both "subscribers" a port with separate VLAN, then correctly firewall between them. This actually would be pretty secure for everyone, and more closely matches what an ISP would (ideally) do.
In addition, he doesn't want a centralized logging of network traffic like most home modem and routers do.
This is a BIG problem, if your tenant does not trust you to be their ISP, you can not be their ISP. In the above scenario with an expensive business router, they still need to trust you. Most carriers will gladly provide two independent services to an address, it is the only "zero-trust" solution available.
Trust goes the other way too, like if your tenant downloads a million movies, or does crime, or violates the ToS of your internet service, it will likely come back to you!