The discovery of the Higgs boson is not verification of the Higgs mechanism. The discovery demanded the existence of a spin-0 particle with non-zero couplings to the Standard Model. For the Higgs mechanism, we must establish that the vacuum of the Higgs potential spontaneously broke electroweak symmetry and generated mass for fermions, gauge bosons, the Higgs boson itself.
To verify the Higgs mechanism, we must measure the cubic and quartic Higgs self-interactions, thus reconstructing the entire Higgs potential, and precisely measure the Higgs interactions with fermions and bosons. This is summarized by the introduction of e.g., 1510.07697,
The discovery in 2012 [1, 2] of a Higgs boson exhibiting properties
similar to those expected from the Standard Model 3 has been one
of the most important developments of the last decade in
experimental particle physics ... This observation however
only consists of the first ingredient allowing one to establish the
Brout-Englert-Higgs mechanism and to fully confirm the Standard Model
nature of the observed new state. Any conclusive statement indeed
requires, in addition to the information currently available from
experimental data, at least a more detailed knowledge of the form of
the Higgs potential.
The Higgs mechanism predicts that interaction strength with Higgs should be proportional to mass, and thus far we find good agreement, as indicated below - according to the Higgs mechanism, the data should lie on a straight line.
Measuring the Higgs self-interactions, on the other hand, is very challenging with current experiments. See e.g, 1206.5001. Their measurement would allow us to check the Mexican hat shape of the Higgs potential.
We're more certain that the gauge bosons obtain their masses from the Higgs mechanism (or a Higgs-like mechanism). The Higgs interactions with gauge bosons are fixed by gauge symmetry (unlike the Yukawa couplings that give rise to fermion masses) and thus we know them from measurements of the gauge couplings in other processes. The ratio of the measured gauge boson masses, $M_W / M_Z$, is in good agreement with predictions of the Higgs mechanism. Technically, this agreement is quantified by the $\rho$-parameter,
$$
\rho = \frac{M_W^2}{M_Z^2 \cos^2\theta_W}
$$
which is measured to be almost exactly one, and predicted (at tree-level) to be one in the Standard Model Higgs mechanism (and similar mechanims with multiple doublets or particular relations between isospin and hypercharge).
