The Higgs boson represent an exception, and is the result of quite a complicated mechanism. And if the new particle were a Higgs-like boson it would be even more surprising.
Except for the Higgs boson, the usual picture is that matter particles are fermions and force mediators are bosons:
One interpretation I've seen is that there are 4 Higgs bosons, three of which are "swallowed" by 3 of the the weak-force bosons which and become the W+ W- and Z. The remaining 4th Higgs boson (the 4th degree of freedom of the Higgs field) is what we can see, together with the 4th weak-force boson, the photon, which remains massless.
So I think it's reasonable to consider that the Higgs boson belongs to the weak force.
I'm not a physicist and I barely understand the ELI5 version of this stuff, but no, I do not think that saying that the Higgs 'belongs' to the weak force is correct as it is a proper excitation of a separate fundamental field.
IIRC, the Higgs field is a scalar field (same field strength at all points of space/time), whereas the other fundamental forces are vector fields (field strength varies with space/time).
Is this correct? And if so, does this fact matter in deciding whether to say Higgs == fundamental force?
It depends on the point of space-time, but at each point of space-time it is just a scalar value, unlike a vector for a vector field. Moreover in physics we classify fields by their Lorentz transformation properties/representation. So when we say scalar field we actually mean that is transforms in the "(0,0) representation" of the Lorentz group (although then it's not yet specified whether one means scalar or pseudoscalar).
If it were the same at all space time points it would be a constant, and rather boring (more like a cosmological constant).
AFAIK, in general relativity, the gravitational field is not a vector field but a tensor field.
EDIT: btw, you got the scalar/vector definition wrong.
A scalar field is one where each point in space can be labeled by a single scalar (not the same vale for all points), for example a temperature reading at different locations.
A vector field assign a distinct vector (i.e. magnitude and direction) to each point in space, for example if you take a compass around the globe you can assign to each location the direction the compass arrow is pointing.
A tensor field is of course assign a tensor to every point in space. I only have rough understanding of what a tensor is, as a first approximation it can be understood as a matrix with special properties.
A false sillogism, indeed, but that's how you make headlines. And btw doesn't make the discovery any less exciting.