Is "gateway" a concept between two or more networks or a concept belonging to just one network?

When a device is connected to two networks, if it is listed as gateway in the routing tables of the hosts in one network, must it be not listed as gateway in the routing tables of the hosts in the other network?

Does a gateway device always have to be connected to a network where it is not a gateway? (So that the default routing table entry can route traffic to the network?)

Can a device be the gateway for each of the networks that it is connected to? If yes, is it correct that:

• the default entry in its routing table will route to a network where it is a gateway?

• will that cause infinite-loop for routing for a destination address which is not in any network that it is connected to and therefore falls into the default entry of its routing table?

Thanks.