Can two networks connected to a router both have a host with the same IP address but a different subnet mask?

Except in very exceptional, non-standard and broken networks the configuration show in the diagram is not valid. (It could be valud if there were 2 routers and 1 switch, and the routers were doing NAT and the devices between the networks never communicated with each other.

The netmasks are almost irrelevant in the diagram above because despite the different netmasks everything would fit in the same /24. The few cases where the netmasks would otherwise change routing decisions are nullified by the router.

In this case, devices in the first network would not be able to see any devices in the second network and vice versa. In fact, the router serves no valid function.

** Ive ignored special cases where the router is deliberately doing non-routerish things like acting as a bridging firewall, and doing dodgy arp / ip manipulation stuff which is extremely non-standard and something most routers cant do. Even then, it is impossible for 2 devices with (only) the same IP addresses to communicate with each other.

** PCs will not send packets to the router where they are destined for hosts in the same subnet.