It doesn't look exactly like the Nano, but is the port named GND same for both?
Looks like it from the pictures.
Why should I connect the GND to each other btw?
A circuit (loop) is necessary for electricity to flow. TX (connected to RX) and GND form a circuit for sending data from one side to the other. RX (connected to TX) and GND form a circuit for sending data the other direction. Without GND, there is no circuit and no electricity flow.
Should I do the same with the 5V?
If you aren't now then you should not. You would only connect 5Vs together if you have a single shared power supply.