How to use two Ethernet connections at the same time (on single pc) on Ubuntu 21.04? [duplicate]

From ifconfig and netstat outputs you provided, it looks that both your connections are assigned IP addresses from the same network - 169.254.0.0 with netmask 255.255.0.0. That may be the reason why they can't be used simultaneously.

The network 169.254.0.0 is the so called autoconfiguration network. It is used when you don't specify manually any IP address in network settings, but leave the setting at default automatic assignment and there is no DHCP server in the network. In that case the connection gets a randomly selected IP address from that network. Thus, computers connected to this network can talk to each other without further configuration, but the downside is that you can have only one such network. You can't connect two autoconfiguration networks to two different network cards on one computer.

Instead of using the autoconfiguration network, you should manually configure IP addresses on all your PCs so that you have two clearly different networks.

For example, you can assign the network card enp6s0 the address 192.168.6.1 with netmask 255.255.255.0, and the network card enp4s0 the address 192.168.4.1 with netmask 255.255.255.0. They are on different networks now. Respectively, you should set the address 192.168.6.2 (with the same netmask) on the remote desktop PC, and the address 192.168.4.2 on the file transfer PC.

After you set this, your PC should be able to communicate with each of the two other PCs, but those two PCs will not be able to communicate with each other. If you need this too, you should enable IP forwarding in the kernel on your PC, so that it will act as a router and transmit packets between the two networks. To enable IP forwarding, you should uncomment the line net.ipv4.ip_forward=1 in the /etc/sysctl.conf file (or add it to the file if it isn't there), and then apply the changes with sudo sysctl -p. Also adding explicit routes to the other network on both remote PC and file transfer PC will probably be necessary.