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. 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. 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. 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 73pm 20 for CO$^+$ in comet C/2016 R2. This value is
in agreement with the Solar System ratio of 89pm 2 within the error bars, but is
also consistent with the 12C/13C ratio in local interstellar medium (68pm 15).