Indeed, N=2 is much more stringent than N=1, and, significantly, it contains a central charge. An N=1 scalar multiplet is CPT self-conjugate, and consists of one Majorana spinor, a scalar, and a pseudoscalar, so (-1/2,0,0,1/2). You might, very naively, think that with two supersymmetry generators available, you could still transition from the spin 1/2 state to either the scalar or the pseudo scalar, each with a different susy generator, and then back to the spin -1/2 fermion. But it turns out this is impossible, so you need two such scalar multiplets, rotating into each other by the peculiar central charge of that algebra (N=2).
Fayet (1976), "Fermi-bose hypersymmetry." NuclPhys B113 135-155 in his hyper symmetry paper, saw that, so it would lead to a complex structure instead of relying on it; but if his language were unclear, to see the doubling is actually inevitable in real variables, without assuming anything,
you may try the two-liner brute-force proof on pp 15-16 of Zachos' 1979 dissertation.
In essence, CPT and hermiticity conspire to require "more room" for the relevant Pascal triangle.
The requirement of two Majorana spinors, two scalars, and two pseudoscalars to realize N=2 is best evident in the two supercurrents (2.1-8) of that second reference: it is manifest that just one spinor and one scalar can only specify one, not two, distinct supercurrents. You must tack together two N=1 scalar supermultiplets to get the N=2 variant.