Error message in linux's dmesg: eno1: Detected Hardware Unit Hang

Answers:

Could somebody tell me whats that error means?

The complete answer on this question requires the deep knowledge in several disciplines: digital hardware design, system programming, linux kernel design and some other. The simple variant of the answer can be sound like 'something has going wrong while the NIC was transmitting the packet'

Is it serious problem or not really?

This error causes the network disconnect on your PC.

Or how to diagnose it?

You already know how to diagnose it - by checking the logs (commands dmesg or better journalctl -k)

Could that be related with my network? Seems that my router also got stuck in the same time.

This issue related only your linux system. I think your router is fine. It's your PC losts connection to network, include the connectivity with the router.

This is a known issue. Almost always it's a driver related problem. You can try to install the e1000e driver from the intel site or update the kernel. If it won't help try to disable the offloads (details in the man ethtool). Otherwise use the other network adapter (based on intel i350 NIC).