There are a few ways to test that. Such as magnetic moment of the particle. When proton's magnetic moment was measured to differ a lot from the QED prediction for an elementary particle, it was a dead giveaway it's composite. OTOH, electron's measured magnetic moment matches QED perfectly.
How Do You Tell If A Boson Made Out Of Component Bosons Is Composite?
There is certainly no positive evidence to date that the Higgs boson is composite, and certainly not that it is more composite than any other Standard Model fundamental particle.
But, my intuition is that it is harder to determine experimentally that a boson, which is composed of other bosons, is composite than it is to determine that a fermion is, because Fermi exclusion which gives rise to spatial separation of the fermionic components of the composite particle, is one important tool used to test compositeness.
Am I wrong?
For example, the physical separation of the charges of the component fermion quarks is why the magnetic moment of a proton is different from the magnetic moment of a positron with the same electric charge.
But, to give a concrete example, suppose that the Higgs boson was a bound state of a W+ boson, a W- boson, a Z boson and a photon superimposed upon each other in exactly the same point in space-time, whose mass is somewhat different than the sum of the rest masses of the W+, W- and Z bosons due to the bound energy of the photon and/or any other binding energy and/or more fundamental Higgs boson kernel particle(s) involved.
What could you observe about a Higgs boson to know that the Higgs boson was composite in that case?
Alternately, would the example be inconsistent with what is meant by a composite Higgs boson in some respect?
Are We Suffering From A Category Error?
Consider the case where the observable properties predicted for a particular type of "composite particle" has no observable differences from a truly fundamental particle. Is calling it either a composite or a fundamental particle an inaccurate thing to do?
I think that it is possible that we have reached a point where the "composite v. fundamental" dichotomy may be a category error.
At the bottom of it all, everything is made of "matter-energy" which is strictly conserved in the SM, so there is that deeper "stuff" which we know can be converted into other things made out of the same "stuff" so long as baryon number, lepton number, electromagnetic charge, spin and maybe another quantum number or two that I'm overlooking are conserved (with the bonus feature that each of conserved quantities, in counter-balancing pairs, can be created from "nothing" but matter-energy, and can annihilate into "nothing" but the same amount of matter-energy with none of the "conserved" quantity behind).
For example, a Higgs boson, which has no electric charge and no color charge and no baryon number, and has spin-0, can (and indeed, usually does) decay to a quark and anti-quark pair, with each of the resulting particles having electric charge, any of six possible color charges (counting color charges and anti-color charges as different), spin, and baryon number, in each case in opposite directions that cancel each other out, allowing conserved quantum quantities to be conserved.
When a Higgs boson decays the resulting products of the decay generally have much great "structure" in terms of quantum numbers spread apart from each other in space-time than the source Higgs boson does, yet the Higgs boson forms that greater structure spontaneously and almost instantly.
To use a quite possibly flawed analogy, the recipe for every Standard Model particle, with the possible exceptions of the top quark, the neutrinos, and any BSM particles , is contained in every Higgs boson.
All "fundamental" particles in the Standard Model can be transformed into other SM "fundamental" particles via Standard Model processes that conserve matter-energy.
A particle that is known to decay into different particles may be fundamental in a narrow technical sense particular to fundamental physics, but in ordinary sloppy day to day English, we don't think of ordinary things that can break into, or transform into, different things to be "fundamental". Even if the SM particles have no intermediate structure short of the basic stuff of matter-energy, and hence are not "composite" because the are structures assembled from other distinct parts, that doesn't really quite imply that they are fundamental units in a layman's sense of the word either.
An absence of compositeness does not inherently mean that so called "fundamental" particles could not be configurations of fundamental "stuff". For example, in (some versions of) string theory, all "stuff" that has matter or energy in the universe is at root a finite tiny string that is not actually truly point-like, and each of the different fundamental particles is a different string resonances of that one fundamental kind of string.
For example, if a Higgs boson and a top quark are both particular kind of excited states of individual strings, then they aren't truly fundamental even though they are also not "composite", because they each have only a single piece, and indeed, the same underlying piece whose properties would determine the range of possible particles in the universe.
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