Lets answer the most important question first, makes explaining the rest much easier.
Is it possible that two devices have the same IP-address
No. In a network, you can never have 2 devices that have the same IP address. If this happens, you have an IP conflict, and one of the two devices will lose network connectivity, possibly both.
If I connect two routers via ethernet, I can have two W-LANS.
You can, yes. Or you can let both send out the same SSID and password, so you only have to configure your wifi once, and it connects to the closest router automatically.
So would I also have two DHCP-Servers
If you don't disable the DHCP server of one of the routers, yes, you would have two DHCP-Servers.
Or use one as a DHCP-Server and the other should have some other functionality besides DHCP?
Keep in mind that a router creates a subnetwork. If you connect router1's LAN to router2's WAN, router2's network is completely isolated from router1. If you want one large network, but want to use the multiple ports of router2, connect router1's LAN to router2's LAN. This disables most of the router parts in router2. You also want to ensure that the DHCP server is disabled too.
I know that globally something like NAT is implemented allows for the same IP-Address for many devices with package request via ports. How is this done locally? Can it be done?
No, what NAT does, is forward any incoming connection from the WAN IP to one of the LAN connected devices. The WAN ip, the ip address given by your ISP is also unique in this network, but as you have a router, that creates a new sub-network, both networks are separated.
If you were to actually put router 2's WAN port in router1's LAN port, router2 would create a new network, with its own set of IP's. You'd create a small version of the internet in your home. As a result, computers in router2's network can't talk to computers in router1's network unless you open ports, and then only small parts talk to eachother. Usually for a home network, this is not what you want unless you need separate networks, for example, a home landlord managing internet, splitting the connection to multiple households.
