Is there any possibilities that two devices have the same public IP?
Most definitely a possibility. There a several possible causes for it to happen, including malicious actors trying to intercept you data.
How is it possible to make sure I am actually going to the right one?
Without some security measures you cannot be sure you are connected to the "right" one. It's even possible that you could be connected to the "wrong" one, and it is connected to the "right" one, and passing on the requested data, after peeking at it, or changing it.
I am not a security expert, so my input on that is highly questionable. My first choice would be to use SSL/TLS with a good key, under your control, at both ends. Without knowing what you are doing while connected, I can't get much deeper anyway. Obfuscation, using some non-standard port, might help some, though it's usually considered very poor as a legitimate security measure.
The keys don't need to be commercial grade, or even have a chain of authority recognized by web browsers. Unless the "server" is intended to be accessed by others. You can tell your browser to "accept" any certificate, including self-signed ones.