Is it possible to have two simultaneous networking connections (one over Ethernet and other over UART using UART to Ethernet adapter)? I also want to know if these two connections can have different IP addresses over the network.
Is it possible to have two simultaneous networking connections
Yes, but you need to have different addresses for the different interfaces (which answers your second question).
However, you probably want to set up routing if you want this to work for any particular purpose. Questions about that are better off on Unix & Linux; make sure you are clear about specifically what it is you want to do beyond just connecting to the same subnet twice (or two different subnets, as the case may be).