  We determine the two-body contact in a planar Bose gas confined by a
transverse harmonic potential, using the nonperturbative functional
renormalization group. We use the three-dimensional thermodynamic definition of
the contact where the latter is related to the derivation of the pressure of
the quasi-two-dimensional system with respect to the three-dimensional
scattering length of the bosons. Without any free parameter, we find a
remarkable agreement with the experimental data of Zou {\it et al.} [Nat. Comm.
{\bf 12}, 760 (2021)] from low to high temperatures, including the vicinity of
the Berezinskii-Kosterlitz-Thouless transition. We also show that the
short-distance behavior of the pair distribution function and the high-momentum
behavior of the momentum distribution are determined by two contacts: the
three-dimensional contact for length scales smaller than the characteristic
length $\ell_z=\sqrt{\hbar/m\omega_z}$ of the harmonic potential and, for
length scales larger than $\ell_z$, an effective two-dimensional contact,
related to the three-dimensional one by a geometric factor depending on
$\ell_z$.
