My question is quite simple: can you, somehow, upload Arduino code without closing puTTY's serial console? I hate having to close the program to open it again for reading the incoming information.
You can't. Putty has no way of disconnecting without closing the program (as far as I can tell).
You could try "Tera Term" which apparently (though I don't use Windows so have no experience with it) has a disconnect option (which you would have to use before uploading).
Alternatively, as @Juraj has mentioned, you could use a separate USB to TTL UART adaptor to provide a second debug serial port (either with SoftwareSerial or mirroring the TX pin of the Arduino directly - don't connect RX in this case) which doesn't interfere with software upload.
There is an open source project that will do what you want.
The com0com will create virtual com ports for the programs to connect to.
The hub4com will allow you to route data between the physical port and multiple virtual com ports.