Let's use your parenthetical as an excuse to keep charge vanishingly small, because we can avoid thinking "Which charge? Which charge carrier or carriers? What's the distribution of charges?", and largely ignore the electro- effects of electrovacuum (which answers these, but in surprising ways when you look deeply).
In a chargeless vacuum Schwarzschild or Kerr universe, we have total coordinate freedom because there is nothing there but the mass at an infintesimally small point, p. If one builds a system of coordinates with p always at a single spacelike point (say, the spacelike origin), then the symmetries of this vacuum system let one chop away spatial position (e.g. const.coord.x, const.coord.y, const.coord.z, t -> 0,0,0,t) and consequently the vector-quantity linear momenta of the black hole vanish. Moreover, these vacuum spacetimes are also eternal (the black holes do not grow or shrink), so we can do t = const. too. With suitable coordinates, gives us two free parameters: mass & spin (and in Schwarzschild, just one: mass).
These solutions do not superpose additively. By the Raychaudhuri focusing theorem, if we add a point mass to the Schwarzschild black hole universe, the two masses will eventually collide. We have broken the spherical symmetry of the Schwarzschild solution, and when we solve the geodesic equations, we find caustics, where our two infinitesimal masses can be in the same infinitesimal space. We have also broken time symmetry: at past time the two masses are spacelike separated. In the future they are not, as they will merge into one black hole. We also have the problem that the black holes move with respect to one another, so we either adapt or system of coordinates to be comoving with the black holes, or we have one or both of them move against the spatial-coordinate part of our system of coordinates.
When we take this further by breaking other symmetries than the time one, e.g. by adding angular momentum to the system, we have to consider the evolution of orbital angular momentum of the pair, and possibly the spin angular momentum of each. Either of our previous-paragraph choices with respect to encoding the coordinate evolution of the spatial distance between the pair of black holes complicates the calculation of the related vector quantities.
We're still in the land of a small number of parameters, but have gone from the three time-independent [mass, spin, charge] to eleven time-dependent [mass, spin, charge, 4-location, 4-linear momentum].
We can explode the number of parameters though by returning to "what is charge?", and equipping these universes with fields of matter. At this point one runs right into the question of: "does the no-hair conjecture hold in a physically plausible universe surrounding a theoretical black hole?" or almost equivalently "when do theoretical black holes fail to approximate astrophysical black holes?", and Ligo/Virgo are good laboratories for studying whether merging black holes go completely bald.
(The hair that is supposed to bald away may be soft and indirect: the circulation of gas and dust at a distance may reveal that a given black hole was previously more than one black hole. After merger, none of that should make a difference to anything (including a 3rd black hole) falling into the balded merged black hole that the merged black hole was previously two black holes. More critically, in the enormously distant future, the evaporation of the merged black hole should not reveal the number or types of objects that fell into it during its history, whether those are black holes or some neutral mix of standard model particles. The Hawking radiation spectrum at any moment should depend only on the eleven parameters in the previous paragraph. But maybe that still-outside dust and gas has memory that remains relevant arbitrarily far into the future. Or maybe classical general relativity is wrong and rather than being crushed into a memoryless ultramicroscopic point, the ingested gas, dust and other black holes retain or at least reveal their individual identities even during evaporation).
Questions like these make black holes extremely interesting, I think.