We have acquired Hubble Space Telescope (HST) and Very Large Telescope near-infrared spectra and images of supernova (SN) Refsdal after its discovery as an Einstein cross in fall 2014. The HST light curve of SN Refsdal has a shape consistent with the distinctive, slowly rising light curves of SN 1987A-like SNe, and we find strong evidence for a broad Hα P-Cygni profile and Na I D absorption in the HST grism spectrum at the redshift (z = 1.49) of the spiral host galaxy. SNe IIn, largely powered by circumstellar interaction, could provide a good match to the light curve of SN Refsdal, but the spectrum of a SN IIn would not show broad and strong Hα and Na I D absorption. From the grism spectrum, we measure an Hα expansion velocity consistent with those of SN 1987A-like SNe at a similar phase. The luminosity, evolution, and Gaussian profile of the Hα emission of the WFC3 and X-shooter spectra, separated by ∼2.5 months in the rest frame, provide additional evidence that supports the SN 1987A-like classification. In comparison with other examples of SN 1987A-like SNe, photometry of SN Refsdal favors bluer B - V and V - R colors and one of the largest luminosities for the assumed range of potential magnifications. The evolution of the light curve at late times will provide additional evidence about the potential existence of any substantial circumstellar material. Using MOSFIRE and X-shooter spectra, we estimate a subsolar host-galaxy metallicity (8.3 ± 0.1 dex and <8.4 dex, respectively) near the explosion site.