  Tensor-network methods are used to perform a comparative study of the
two-dimensional classical Heisenberg and $\mathrm{RP}^2$ models. We demonstrate
that uniform matrix product states (MPS) with explicit $\mathrm{SO}(3)$
symmetry can probe correlation lengths up to $\mathcal{O}(10^3)$ sites
accurately, and we study the scaling of entanglement entropy and universal
features of MPS entanglement spectra. For the Heisenberg model, we find no
signs of a finite-temperature phase transition, supporting the scenario of
asymptotic freedom. For the $\mathrm{RP}^2$ model we observe an abrupt onset of
scaling behaviour, consistent with hints of a finite-temperature phase
transition reported in previous studies. A careful analysis of the softening of
the correlation length divergence, the scaling of the entanglement entropy and
the MPS entanglement spectra shows that our results are inconsistent with true
criticality, but are rather in agreement with the scenario of a crossover to a
pseudo-critical region which exhibits strong signatures of nematic
quasi-long-range order at length scales below the true correlation length. Our
results reveal a fundamental difference in scaling behaviour between the
Heisenberg and $\mathrm{RP}^2$ models: Whereas the emergence of scaling in the
former shifts to zero temperature if the bond dimension is increased, it occurs
at a finite bond-dimension independent crossover temperature in the latter.
