To which of the four forces of nature does the coupling of the Higgs scalar field to other quantum fields ultimately belong to, if any? If it doesn't fall into any of the four known categories of forces, does that imply a fifth "force" of nature, or have I misunderstood the definition of force?
The fields in the Standard model can be put into three categories:
Matter fields (quarks, leptons).
a) Massless at the 'fundamental' level. Massive after aquireing mass via Higgs mechanism.
c) Interact with each other only via other fields, e.g.
Mediators of the interactions (gauge bosons). These are responsible for the presence of the 'forces'.
a) Massless at the 'fundamental' level. W&Z bosons become massive after aquireing mass via Higgs mechanism. Photons and gluons are massless.
c) Couples (interacts with) directly to the matter fields, providing them with masses. This type of interaction is named after Yukawa.
d) Couples to electroweak gauge bosons, providing W&Z with masses.
Now you see how special the Higgs field is. It is a boson which is not a mediator of interactions. It couples (directly!) to both matter fields and electroweak bosons. Typically people refer to d) as to the Higgs mechanism.
Interactions in e) and d) are typically not called 'forces' because they are not due to the presence of gauge bosons; see the discussion here.