51 Eridani b is an exoplanet around a young (20Myr) nearby (29.4pc) F0-type star, which was recently discovered by direct imaging. It is one of the closest direct imaging planets in angular and physical separation (~0.5", ~13AU) and is well suited for spectroscopic analysis using integral field spectrographs. We aim to refine the atmospheric properties of the known giant planet and to constrain the architecture of the system further by searching for additional companions. We used the extreme adaptive optics instrument SPHERE at the Very Large Telescope (VLT) to obtain simultaneous dual-band imaging with IRDIS and integral field spectra with IFS, extending the spectral coverage of the planet to the complete Y- to H-band range and providing additional photometry in the K12-bands (2.11, 2.25 micron). We present the first spectrophotometric measurements in the Y and K bands for the planet and revise its J-band flux to values 40% fainter than previous measurements. Cloudy models with uniform cloud coverage provide a good match to the data. We derive the temperature, radius, surface gravity, metallicity, and cloud sedimentation parameter fsed. We find that the atmosphere is highly super-solar ([Fe/H]~1.0), and the low fsed~1.26 value is indicative of a vertically extended, optically thick cloud cover with small sized particles. The model radius and surface gravity estimates suggest higher planetary masses of M_gravity_=9.1^+4.9^_-3.3_. The evolutionary model only provides a lower mass limit of >2M_jupiter_ (for pure hot-start). The cold-start model cannot explain the luminosity of the planet. The SPHERE and NACO/SAM detection limits probe the 51 Eri system at solar system scales and exclude brown-dwarf companions more massive than 20M_jupiter_ beyond separations of ~2.5AU and giant planets more massive than 2M_jupiter_ beyond 9 au.