Bomber killers, but total dogs in the air due to the heavy armament.
API when the muzzle velocity was so weak compared to Allied versions isn't going to be as good as HEI-T shells. I'm not sure how they could fit something so long, heavy, and large in the Bf109's nose.
Soviets went even further, installing the full-power, automatic 37mm cannon aboard the LaGG-3 and Yak-9.
And the Soviets installed necked-out 45 mm NS-45 cannons and experimented with 57 mm cannons in the Yak-9s propeller hub. The 57 mm cannons were not successful because the barrel was too short to extend past the cylinder banks, so the muzzle blast essentially destroyed the engine.
Going back a bit, since our starting point is in 1936 and given that the French were already working on 9 and 11mm HMGs as intermediate anti-material HMGs I don't see why Germany couldn't have simply scaled up the MG17 to something like a 10.75mm bullet (already a caliber in production with Mauser since the early 1920s) with a larger case behind it than already in production to give it something like a 850-900m/s muzzle velocity. You'd have a high rate of fire and ability to fire API and HEI-T rounds that aren't so heavy and complex as the MG131 project and be light enough to fit a bunch of them in even a platform as light as the Bf109. The penetration of a long AP round should be excellent at 850-900m/s due to sectional density, which is why the French were looking at them as ground based anti-material weapons without going up to 13mm caliber HMGs with the weight they'd bring, while being wide enough to fit in sufficient special loads.
That would work, but it should be noted that the MG131 was already close to a scaled-up MG17. It was designed by Louis Stange like the MG30 (the basis for the MG15, MG17, MG81, and MG34), used the Solothurn Lock like the MG30 family did, and in general had similar design. Details were different, though, with electrical firing and other minor differences.
One other thing that should be noted in general is that a gun's rate of fire will slow somewhat as the gun's mechanism is scaled up. The power of the recoil (or gas piston) increases due to a heavier shell, and this greater power should drive the mechanism at a higher speed, but it is more than canceled out by the other increases. The size and mass of both the cartridge and bolt increase, requiring more force to move them at a given speed, and since the cartridge is longer, it will have to move back and forward a greater distance to eject and reload the rounds. Both of these mean that gun rate of fire tends to decrease as a mechanism is scaled up, even though the Q factor usually remains the same at a given muzzle velocity (it is after all a measurement of the inherent mechanism of the gun, not its size).
On the WWII aircraft gun chart, it can be seen that guns of a given family have decreasing rates of fire as the cartridge power increases. The most prolific gun family on the chart is the Browning, with the .30 M2 and .303 having 1150-1200 rpm, the Ho-103 having 800-900 rpm, the .50 M2 and Ho-5 having 750-850 rpm, the Ho-155 having 400 and then 500 rpm, and the Ho-204 having 300-400 rpm. These all used the same basic mechanism. The same can be seen with the Oerlikon family (including the MG FF and 20 mm Type 2), the Berlin UB and B-20, the ShKAS and ShVAK, the NS-37/45 and NS-23, etc.
For this reason the challenge for aircraft gun (and antiaircraft gun) designers is to build a mechanism with the maximum possible Q factor. In other words, a mechanism that can handle large rounds with high muzzle energy and still cycle them quickly (high rate of fire) for little weight.
The Russians did something similar with a 30mm cannon:
https://en.wikipedia.org/wiki/Gryazev-Shipunov_GSh-30-1
Same recoil system and higher rate of fire, though a bit more complex due to extra features.
In principle of operation yes, they're both short recoil, but the mechanism is very different, and actually simpler than many WWII aircraft cannons. The specific design of that mechanism is what gives the cannon its very high rate of fire for a linear action gun of that power.