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 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 +/- 5.0 d, 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 SN Ia host galaxy at z = 0.4087, as well as in the foreground lensing galaxy at z = 0.2163. We detect strong Naid absorption in the host galaxy, indicating that iPTF16geu belongs to a subclass of SNe Ia displaying 'anomalously' large Naid column densities compared to dust extinction derived from light curves. For the lens galaxy, we refine the measurement of the velocity dispersion, sigma = 129 +/- 4 kms(-1), which significantly constrains the lens model. We use ground-based spectroscopy, boosted by a factor similar to 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 is a normal SN Ia. 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.