The $\eta$ Carinae binary system hosts one of the most massive stars, which features the highest known mass-loss rate. This dense wind encounters the much faster wind expelled by the stellar companion, dissipating mechanical energy in the shock, where particles can be accelerated up to relativistic energies and subsequently produce very-high-energy $\gamma$-rays. We used data from the Fermi Large Area Telescope obtained during the first 7 years and spanning two passages of $\eta$ Carinae at periastron and compared them with the predictions of particle acceleration in hydrodynamic simulations. Two emission components can be distinguished. The low-energy component cuts off below 10 GeV and its flux, modulated by the orbital motion, varies by a factor less than 2. Short-term variability occurs at periastron. The flux of the high-energy component varies by a factor 3-4 but differently during the two periastrons. The variabilities observed at low energy, including some details of them, and these observed at high energy during the first half of the observations, match the prediction of the simulation, assuming a surface magnetic field in the range 0.4-1 kG. The high-energy component and the thermal X-ray emission were weaker than expected around the second periastron suggesting a modification of the wind density in the inner wind collision zone. Diffuse shock acceleration in the complex geometry of the wind collision zone of $\eta$ Carinae provides a convincing match to the observations and new diagnostic tools to probe the geometry and energetics of the system. Orbital modulations of the high-energy component can be distinguished from these of photo absorption by the four large size telescopes of the Cherenkov Telescope Array to be placed in the southern hemisphere. e-Astrogam could easily discriminate between the lepto-hadronic and the hadronic models for the gamma-ray emission and constrain acceleration physics in more extreme conditions than found in SNR.