RPi5 with LinuxCNC Image and Waveshare PCIe TO Gigabit ETH Board

This is the output of ip a

cnc@linuxcnc1:~$ ip a
1: lo: <LOOPBACK,UP,LOWER_UP> mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000
    link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
    inet 127.0.0.1/8 scope host lo
       valid_lft forever preferred_lft forever
    inet6 ::1/128 scope host noprefixroute
       valid_lft forever preferred_lft forever
2: eth0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc pfifo_fast state UP group default qlen 1000
    link/ether 2c:cf:67:3a:15:bb brd ff:ff:ff:ff:ff:ff
    altname end0
    inet 192.168.0.153/24 brd 192.168.0.255 scope global dynamic eth0
       valid_lft 799842sec preferred_lft 799842sec
    inet6 fdce:a2be:2d61:0:2ecf:67ff:fe3a:15bb/64 scope global dynamic mngtmpaddr
       valid_lft 7124sec preferred_lft 3524sec
    inet6 2003:e4:9741:1e00:2ecf:67ff:fe3a:15bb/64 scope global dynamic mngtmpaddr
       valid_lft 7124sec preferred_lft 1191sec
    inet6 fe80::2ecf:67ff:fe3a:15bb/64 scope link
       valid_lft forever preferred_lft forever
3: eth1: <NO-CARRIER,BROADCAST,MULTICAST,UP> mtu 1500 qdisc pfifo_fast state DOWN group default qlen 1000
    link/ether aa:57:f1:85:a9:88 brd ff:ff:ff:ff:ff:ff
    altname enp1s0
4: wlan0: <NO-CARRIER,BROADCAST,MULTICAST,UP> mtu 1500 qdisc pfifo_fast state DOWN group default qlen 1000
    link/ether 2c:cf:67:3a:15:bc brd ff:ff:ff:ff:ff:ff

When connecting the eth1 port to my network switch I got an information from my router, that it assigned a new IP adress to the ethernet card with the MAC adress of eth1. So I assumed that it should be possible to reach that port via ping. This ping was done from a windows PC in my network. All four tries timed out. But looking to the result of ip a above, the IP adress is not shown on Linux side. This I did not see before.

The reason why I would like to use this second ethernet is because my workshop is in the basement of my house and the RaspberryPi is in a metal housing within a metal cabinet. Finally I like to use my devices hardwired instead of using WIFI. I thought it would be straight forward to put in an additional ethernet card, activate PCIe and then be able to use it.

Unfortunatelly I am not an Linux expert and I am not a native english speaker, so if I miss something from your advices please give me a hint. I will try to make it. 

You need to go into gnome network manager and configure that interface, even if it just to get a dhcpd address.

I suggest you look up some information from Debian in your own language to find out more regarding networking.
This forum is regarding topics primarily related to Linuxcnc, whilst some generally Linux support is given you seem to be at a level where you require some more intensive help.
The RPi recognises the card and loads the driver, the kernel has created eth1 related to the NIC, the hardwork is done, now you need to find some more in depth knowledge regarding networking. As I said before the Linuxcnc image is basically Debian but with a realtime kernel built from the RPi GitHub repo and Linuxcnc installed.