Obviously, the possible numbers if intersections for, one branch of a hyperbola and one branch of another hyperbola, are: $0, 1, 2, 4$ (check the example here). Is it possible for $3$?
If the two branches share the same focus, what's the maximum number of intersections?
The only way to get three intersection points is if one of them is a "double" intersection (i.e. a point where the two curves are tangent to one another).
To see why ...
You can write one curve in parametric form $\mathbf{X} = \mathbf{X}(t)$, using rational quadratic functions, and the other in implicit form $f(\mathbf{X}) =0$, then intersections occur at real values of $t$ that satisfy $f(\mathbf{X}(t)) =0$. This leads to a polynomial equation of degree $4$. This equation generically has either 0, 2, or 4 real roots. If there are two real roots, and they are equal, then the two curves are tangent at some point, and you might say that there is only one intersection. If there are four real roots, and two of them coincide, then you might say that there are three intersections.