Can I run 2 Bitcoin nodes with the same wallets? We have a full Bitcoin node with its wallets on the server. The issue is that the majority of our wallets were synchronized long time ago. For us, it would seem that this synchronization task can take a significant amount of time in which the Bitcoin node will be unreachable for all other requests and our server will be unavailable for users.

We think about replicating the current Bitcoin node on the separate server and synchronize wallets there. After this process will be finished we will copy wallets back to the origin server. 

Is it doable? Or we can't have the same wallets on two different Bitcoin nodes?
This is doable. You can run as many nodes with the same wallets as you want.
I imagine you want to do this because the synchronisation process will be faster on another node?
After the process is done, you will have to copy the complete blockchain data, not the wallet itself.

You might want to consider checking for what your bottleneck is - could be the hard disk or the CPU, or in some cases the internet bandwidth.