As this is a cupboard door going into what looks like a door casing then concealed (kitchen door) hinges are far from suitable- you really need to go for something like butt hinges. Whatever material you use for the door frame, its' thickness must be great enough to accommodate the hinges. I'd therefore start by sourcing some appropriate size butt hinges
Because you are not constrained to the 15 to 22mm thickness range that the vast majority of concealed hinges can accommodate your door frame can be thicker. I'd suggest that you should consider something like 30 to 35mm thick timber for the door frame (I.e a similar thickness to a thin domestic door). This would allow you to groove the inside of the frame components to take a panel section thick enough to have the tongue and groove pattern you desire. The door frame would be assembled in a U-shape with one stile missing and thin match board (tongue and groove) pieces would be slid in to form your panel. Finally the second stile would be fixed in place.
This approach requires a router and a bearing guided grooving cutter of appropriate thickness for your match boards and with a depth of cut of 6 to 10mm. I recommend that your panel section is made up by glueing the t&g boards together on the bench as a single section then trimming to size before they are inserted into the frame. Solid wood would allow you to stain (it will need to be stained, I'd say) then oil it to the required shade to match the door
As an alternative you might want to consider using a routed MDF panel. Instead of t&g matchboard you make up a single MDF panel to go into the door. To get the T&G look a number of parallel v-groove cuts are made using a router, v-groove cutter, straight batten (as a guide) and a pair of G- cramps (or similar). This approach would only be suitable for a painted finish (or possibly a stained and oiled door frame with a painted centre panel
Notes
If using solid wood allow it to acclimatise in your house for 3 to 5 days before starting the project.
You should NOT glue the panel section in place and you must leave a bit of room (a couple of millimetres off both height and width) to allow for wood movement.
It is far better to pre-finish your panel section before installing it this avoiding a white witness mark at the edges should the frame shrink slightly (which can happen in centrally heated houses)
It looks as though you will need to add a stop lsth inside the opening
An alternative approach is to make the frame up, put a rebate in the back (router, bearing guided rebate cutter), square the corners of the rebates (sharp chisel), install your panel.into the rebate then secure it with a small profile beading all round, something like a quarter round profile with mitred corners (saw, mitre block, rampin)