We report the results from spectroscopic observations of the multiple images of the strongly lensed Type Ia supernova (SN Ia), iPTF16geu, obtained with ground based telescopes and the Hubble Space Telescope (HST). From a single epoch of slitless spectroscopy with HST, we can resolve spectra of individual lensed supernova images for the first time. This allows us to perform an independent measurement of the time-delay between the two brightest images, $Delta t = 1.4 pm 5.0$ days, which is consistent with the time-delay measured from the light-curves. We also present measurements of narrow emission and absorption lines characterizing the interstellar medium in the host galaxy at z=0.4087, as well as in the foreground lensing galaxy at z=0.2163. We detect strong Na ID absorption in the host galaxy, indicating that iPTF16geu belongs to a subclass of SNe Ia displaying anomalously large Na ID column densities in comparison to the amount of dust extinction derived from their light curves. For the deflecting galaxy, we refine the measurement of the velocity dispersion, $sigma = 129 pm 4$ km/s, which significantly constrains the lens model. Since the time-delay between the SN images is negligible, we can use unresolved ground based spectroscopy, boosted by a factor ~70 from lensing magnification, to study the properties of a high-z SN Ia with unprecedented signal-to-noise ratio. The spectral properties of the supernova, such as pseudo-Equivalent widths of several absorption features and velocities of the Si II-line indicate that iPTF16geu, besides being lensed, is a normal SN Ia, indistinguishable from well-studied ones in the local universe, providing support for the use of SNe Ia in precision cosmology. We do not detect any significant deviations of the SN spectral energy distribution from microlensing of the SN photosphere by stars and compact objects in the lensing galaxy.
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