Is it possible to buy a custom motherboard from AMD with only the specified (by me, the customer) ports?

I would also like to specify the size of the board. Is something like this even possible?

i dont think so. Its highly unlikely that they will create a custom board for a single person. Maybe its possible but it will highly depend on how many boards you order.

but i still say its not possible.

I mean custom board but several thousand units. Are there third party companies that do this besides AMD themselves?

There are thousands and thousands of companies that design, make, subcontract, arrange, manufacture all range of boards you will ever want.

AMD is not of those, they just make the chips...

Yes, there are countless companies producing and/or designing any kind of printed circuit boards. (Just an example from EU for a generalist is eurocircuits, but you can find many others)

Still the big question is what you really want to have.

If you have an own design or you are willing to pay for a completely new design, you can go to some generalist PCB company as I mentioned abowe.

With some small series with modifications for a well known design? You can go probably to the original producer company itself and ask for designing a variant for you. Other than the original company, might run into licensing issues.

Depending on how complicated your wish was you will get a cheap or expensive quotation.
You have to pay the project costs (design, tooling, coordination,...) and of course the variable costs (raw material, components, machine time,...)
Even the biggest brand companies (like samsung or similar) would do this for you, just probably not in the price range or minimal ammount, you would possibly prefer .

If you can find a board what AMD really produced themselves (orvia sucontracting but branded AMD), you might have a chance to ask for a variant. If the origin design is not too old and the have personal capacities, it is not a big deal to do any redesign technically.

There might be rather bureaucratic difficulties with getting to the right decisionmakers. You would have to proove somehow, that your PCBs are the most important for them ($$$ )
But still you would probably have to pre-finance the project. (even more $$$$$$)

Anyway I have seen 30 years old PCB designs getting produced again (12 pieces!!), just becaouse they controlled such critical stuff it was worth to search for the already pensioned designer guy, to help and set up the line. Some of the components were scavenged back for it from the old dead PCBs as there were no suppliers any more.( Enough money can make any system to roll as you wish.)