We study the finite-size behavior of two-dimensional spin-glass models. We consider the ±J model for two different values of the probability of the antiferromagnetic bonds and the model with Gaussian distributed couplings. The analysis of renormalization-group invariant quantities, the overlap susceptibility, and the two-point correlation function confirms that they belong to the same universality class. We analyze in detail the standard finite-size scaling limit in terms of TL(1/ν) in the ±J model. We find that it holds asymptotically. This result is consistent with the low-temperature crossover scenario in which the crossover temperature, which separates the universal high-temperature region from the discrete low-temperature regime, scales as T(c)(L)~L(-θ(S)) with θ(S)≈0.5.