NGC 1052-DF2, an ultra-diffuse galaxy (UDG), has been the subject of intense debate. Its alleged absence of dark matter, and the brightness and number excess of its globular clusters (GCs) at an initially assumed distance of 20 Mpc suggest a new formation channel for UDGs. We present the first systematic spectroscopic analysis of the stellar body and the GCs in this galaxy (six previously known and one newly confirmed member) using MUSE at the VLT. Even though NGC 1052-DF2 does not show any spatially extended emission lines, we report the discovery of three planetary nebulae (PNe). We conduct full spectral fitting on the UDG and the stacked spectra of all the GCs. The UDG’s stellar population is old, 8.9 ± 1.5 Gyr; metal poor, [M/H] = −1.07 ± 0.12; and with little or no α-enrichment. The stacked spectrum of all GCs indicates a similar age of 8.9 ± 1.8 Gyr, but a lower metallicity of [M/H] = −1.63 ± 0.09 and a similarly low α-enrichment. There is no evidence for a variation in age and metallicity in the GC population with the available spectra. The significantly more metal-rich stellar body with respect to its associated GCs, the age of the population, its metallicity, and its α-enrichment are all in line with other dwarf galaxies. NGC 1052-DF2 thus falls on the same empirical mass–metallicity relation as other dwarfs for the full distance range assumed in the literature. We find that both debated distance estimates (13 and 20 Mpc) are similarly likely, given the three discovered PNe.