Context. CO is an abundant species in comets, creating CO+ ion with emission lines that can be observed in the optical spectral range. A good modeling of its fluorescence spectrum is important for a better measurement of the CO+ abundance. Such a species, if abundant enough, can also be used to measure the 12C/13C isotopic ratio. Aims. This study uses the opportunity of a high CO content observed in the comet C/2016 R2 (PanSTARRS), which created bright CO+ emission lines in the optical range, to build and test a new fluorescence model of this species and to measure the 12C/13C isotopic ratio in this chemical species for the first time with ground-based observations. Methods. Thanks to laboratory data and theoretical works available in the scientific literature, we developed a new fluorescence model both for 12CO+ and 13CO+ ions. The 13CO+ model can be used for coadding faint emission lines and to obtain a sufficient signal-to-noise ratio to detect this isotopologue. Results. Our fluorescence model provides a good modeling of the 12CO+ emission lines, allowing us to publish revised fluorescence efficiencies. Based on similar transition probabilities for 12CO+ and 13CO+, we derive a 12C/13C isotopic ratio of 73±20 for CO+ in comet C/2016 R2. This value is in agreement with the Solar System ratio of 89±2 within the error bars, but is also consistent with the 12C/13C ratio in local interstellar medium (68±15).