Why Ginzburg-Landau theory supposes that superconductivity is a second-order phase transition? According to modern nomenclature or Ehrenfest's one?
Because that what experiments at the time indicated.
For instance, the first (!) line of the 1957 BCS paper states:
The main facts which a theory of superconductivity must explain are (1) a second-order phase transition at the critical temperature, $T_c$ (...)
This is the whole answer.
There is actually a really interesting story here. At the time of Ginzburg-Landau and BCS, everybody was happy with the theories explaining the second-order phase transition. However, the domain of validity of Ginzburg-Landau theory can be derived self-consistently, as explained in many textbooks, or this blogpost. This gives a temperature range, called the Ginzburg temperature, around the critical temperature, within which GL theory can no longer be trusted. This is due to the effect of thermal fluctuations which are neglected in the mean-field theory.
Now for neutral condensates, so superfluids, the Ginzburg temperature interval is quite large, and we now know that the superfluid critical point lies in the 3DXY-universality class, with for instance critical exponent $\beta$ being around 0.35 instead of the mean-field value 0.5. This can be calculated with renormalization group techniques.
A superconductor is not neutral but charged, and coupled to a gauge field, the electromagnetic vector potential. In 1973, Halperin, Lubensky and Ma showed that the fluctuations of the gauge field actually drive the phase transition to become first-order. This is called fluctuation-induced first-order phase transition, and is related to the Coleman-Weinberg phenomenon in particle physics.
However, recalling the point that we started with, experiments show that superconductors behave like a mean-field system: a second-order phase transition with mean-field exponents, to high precision. So it doesn't look like a first-order, neither like a modified (renormalized) second-order transition. The reason is that the Ginzburg temperature interval is very small, and as such, the influence of fluctuations is extremely limited. Most experiments are not sensitive enough to probe the 'true' nature of the phase transition.
(To be nitpicky: only type-I superconductors undergo a fluctuation-induced first-order phase transition. Type-II superconductors' phase transition is still second order, in the 3DXY universality class.)