Except for the Higgs boson, the usual picture is that matter particles are fermions and force mediators are bosons:
https://upload.wikimedia.org/wikipedia/commons/thumb/0/00/St...
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.
Is this correct? And if so, does this fact matter in deciding whether to say Higgs == fundamental force?
If it were the same at all space time points it would be a constant, and rather boring (more like a cosmological constant).
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.