What kind of port/connector is this?

I believe that is a Honda diagnostic port. A "JDM" Japanese vehicle that will never see the states is not required to have a standard OBD2 connector.

I have a Snap-On Solus diagnostic computer, and I believe I have that adapter - there was a time when I had money that I was collecting all of them, regardless if I would ever use it.

Although yours may - some OEMS install an OBD2 port and their own proprietary connector with advanced capabilities beyond the typical OBD mode spectrum. The ability to re-flash the Engine Control Module (ECM/ECU) to new software, for instance.

On Edit:

I now think I'm completely wrong. That (under the AV system) is possibly some sort of HDMI port. I have no idea whether it's an input or an output.