PG 1553 + 113 is one of the few blazars with a convincing quasi-periodic emission in the gamma-ray band. The source is also a very high energy (VHE; >100 GeV) gamma-ray emitter. To better understand its properties and identify the underlying physical processes driving its variability, the MAGIC Collaboration initiated a multiyear, multiwavelength monitoring campaign in 2015 involving the OVRO 40-m and Medicina radio telescopes, REM, KVA, and the MAGIC telescopes, Swift and Fermi satellites, and the WEBT network. The analysis presented in this paper uses data until 2017 and focuses on the characterization of the variability. The gamma-ray data show a (hint of a) periodic signal compatible with literature, but the X-ray and VHE gamma-ray data do not show statistical evidence for a periodic signal. In other bands, the data are compatible with the gamma-ray period, but with a relatively high p-value. The complex connection between the low- and high-energy emission and the non-monochromatic modulation and changes in flux suggests that a simple one-zone model is unable to explain all the variability. Instead, a model including a periodic component along with multiple emission zones is required.