How is a type rating administered for aircraft that do not comply with 91.109?
The short of it is that most such ex-military aircraft, and replicas built new to spec on these designs, do not have a "type" as defined by the FAA at all. Because the absence of a type means the aircraft can't be certified as airworthy in the normal way, and pilots of these aircraft cannot receive a type rating to fly them, the certification of aircraft and pilot is a bit different.

The aircraft must first be issued an FAA "Special Airworthiness Certificate". This is essentially a slightly more rigorous form of airworthiness inspections for type-rated aircraft and virtually identical to that for experimental and/or home-built aircraft, resulting in the certification of the craft as airworthy in general even though it does not fall into a predefined aircraft type for which FAA instructors can issue type ratings.

The airman, because he can't be certified to fly a non-existent FAA aircraft type, must receive a special rating certification called a "Letter of Authorization", allowing the pilot to fly aircraft of a specific model, and/or a "set" of aircraft that have similar configuration, power, weight and other operational specifications and limits (but that aren't similar or common enough for a type rating). The training required for an LOA includes a combination of ground school and flight school containing information and in-flight training on the specific model to be authorized. But, when it is not possible or practical to do in-flight training in the exact model of aircraft to be authorized (usually because the aircraft in question is a single-seater with no two-seat configuration available), flight training in a "similar" aircraft can be a substitute. Pilots of vintage single-seat warplanes like P-51s typically get their flight training for the category in something similar to what military pilots of the day would have used, such as an AT-6. If the flight school just happens to have a Twin Mustang laying around, that would be even better, and also more directly applicable to multi-engine warbirds like the P-38.

Back to your example, say you get your hands on a "de-fanged" A-10 (which would require a ton of ballast weight to be added, literally; the GAU-8 weighs about as much as a Cadillac, but I digress). All flying examples are single-seaters; however, the avionics, especially of the original A-10A without the glass cockpit upgrade of the PEUP program, are comparable to any 70's-era jet trainer. So, you could receive flight training in a T-33, which is a trainer-compatible straight-wing turbojet with more similarities in its flight characteristics to the A-10 than differences. Coupled with ground instruction and/or simulator time on the specific systems of the A-10 (Which won't include any weapons integration, but honestly for the A-10A that was a single FLIR display and the arming/firing switches, and the C variant only upgraded that to a pair of MFDs; other than relative positioning, the flight instruments of the A-10 would be familiar to any PPL), you could be certified to fly your A-10 solo without any flight hours in an A-10 specifically. 

The same might apply to aircraft that do have two-seater variants but where the flight school is unable to procure one. If you won the Powerball, and put a chunk of that money into your very own milsurp F-15C (\$38 million a pop back when they were being made and that's in late 1970s dollars), it's highly unlikely your local flight school would be willing to pony up for an F-15D two-seat configuration to train you on it (unless you bought the plane for them with another big chunk of your lottery winnings). They might, however, have a T-38 in their inventory or be able to loan one from the nearby naval air station, which would give you a primer into the capabilities of a multi-engine turbofan ex-military fighter. Honestly, though, if you wanted to fly an F-15, you'll probably want a two-seater, and your CPL, so you could give rides to paying customers and offset the cost of running this beast (typical fuel load without droptanks, 13,500 lbs which is about 1,985 gallons. With Jet A running about \$4.85/gallon, the cost of a fillup would be somewhere in the $10,000 neighborhood which would give you a cruising range of about 1,000 miles, so you could offer this as a special type of jet taxi).

Here's some more reading on the subject: FAA Flight Systems Information Management System - 8900.1 V5 Chapter 9 Sec 2 - Airman Qualification Requirements for Aircraft for Which the Operating Limitations Require an FAA‑issued Authorization to Act as Pilot in Command