Access home network

I would need to access the home network from external location, in order to have a public internet ip same as the home one. I know about creating a vpn server like with raspberry but I was wondering if it is possible to establish an external connection to the router in order to achieve the same result

No.

Firstly your home router has a firewall which blocks all incoming new connections in the first place, so you need to create a port forwarding rule to even get through that. That rule has to point to a device inside your network, it's not possible to create one that would send the traffic directly back out.

What you can do is run a VNC server on a computer in your home, then use VNC client to access the desktop of the home computer. Then you can use your home computer through the VNC client just the same way as you would sitting at it. The performance will be somewhat laggy, tho - a remote access can never be equal to local use, already laws of physics get in the way.