Why is it important, in terms of ESD protection, to ground a PCB to its case (if not insulative)?
Since your power adapter has no ground/earth connection, there really is no ground connection at all.

Any "ground" of your circuit is just a local ground, it has no relation to mains earth/ground as there is no connection.

You do not have to connect the metal case to your circuit. Note however that if you do connect it to your circuit's (local) ground you will be relying on the mains isolation of the adapter to prevent your metal box from becoming mains live. If for some reason that mains isolation is compromised (for example a fault in the adapter) then your metal box becomes dangerous to touch.

Often it is preferred to use a mains adapter with a ground connection (3 pins at the mains side) and a grounded output meaning the - pole of the DC output is connected to mains earth. Then as soon as you connect the power supply, your metal box is grounded. This is what is used in nearly all professional measurement equipment: metal box which is grounded directly to mains earth.

However, looking at the metal box, it appears that it has a black coating which could provide some protection against direct human contact with the metal.

I think in your case (this mini-PC you're building) you do not need to worry about the grounding, it is quite common to connect the case to the PCB's ground and power that from an adapter without a ground.

Regarding ESD: ESD has little relation to grounding and really you should not worry about ESD. The input and outputs of that (single board computer) PCB will have sufficient ESD protection. As long as you're not deliberately ESD-charging yourself and then discharging yourself through for example USB socket data pin using a metal pin or wire, then you do not have to worry about ESD.

So: just build your project as everyone else would: mount PCB in metal case, power with the adapter you have, stop worrying about ESD.