The reason why say platformers and shooters are compatible with beat em up type games is because they are all sub genres,variations of action games, as in games about controlling an avatar in a games space directly controlling their actions.
So shooters, beat em ups, platformers, and fighting games are all sub genres of action games.
They all ostensibly share the same design but go about it differently. That's why ninja gaiden black has platforming seemelssly integrated into its melee combat, as it's just another action to perform. Stats are from another type of gameplay entirely, that's why turn based combat can have them as it compliments that gameplay mode. Turn based on pen and paper with dice is the same as digital, that's why damage values are given in numbers since it's more methodical mathematic approach not reacting dynamically to physical threats and the environment.
In action games you have various actions being offensive,defensive and traversal it's up to the player to make use of them as best as they can. You could easily slot in a Romanov into ninja gaiden with little to no changes as it's the same fundamental enemy design.
Examples: mega man is a shooter platformer, remove the shooting and it's a platformer, remove the platforming then it's a shooter.
Contra is a shooter with some platforming, with top down and behind the player camera perspectives also in some entries.
Dmc and bayo are melee oriented action games with shooter aspects(guns), they also feature some platforming and minor puzzles.
Castlevania is an action platformer.
This is why action games overall appeal to me and why I avoid anything that isn't about directly controlling an avatar and performing various actions with them. Regardless if we call them action or not or whatever sub genre they may be, they all clearly share the same underlining design philosophy that makes them the way they are and why they are so compatible.
This is why many action games have platforming(bayo, ninja gaiden, dmc) and a shooter sections(viewtiful joe,bayo) as it compliments the core gameplay as it's simply another type of action game to perform. Some emphasise one action over the other, vanquish has melee moves that function complimentary to the shooting. We call it a shooter because that's what you do most of the time, shooting is a primary action of shooters, just as the primary action is platforming in platformers. If vanquish was primarily melee oriented then we would call it a beat em up, god hand has a few projectile moves and you can launch enemies into each other but it's otherwise entirely melee oriented, this is why modern action games like god hand are the evolution of beat em ups, much like how vanquish is the evolution of the run n gun shooter like gunstar heroes and contra. Action games made a drastic leap from 2d to 3D just like platformers did.
Vanquish integrates typical arcade shooter design in the 3D 3rd person cover shooter format, since the shooter action game architecture was always their, just implemented and emphasised differently compared to typical cover shooters which also did have similar action enemy types(gears of war wretches and tickers, shunobi robots and many bosses in binary domain) but relied more on samey restrictive cover systems as opposed to having cover as one facet of gameplay and promoting more movement options like Max Payne and titanfall do.
So that's why beat em ups,shooters and platformers among others can be seemelessly integrated as the framework for them is similar as they all sub genres of action games, all different actions ones game avatar can perform.
the merger between real time strategy and turn based gameplay is more difficult to merge with action games due to how different the gameplay modes are. Something like super hot could be said to be a merger of fps and real time strategy. https://en.m.wikipedia.org/wiki/Action_game
This page should help clear any confusion.
Page for shooters criteria:https://en.m.wikipedia.org/wiki/Shooter_game