I think one cannot say nowadays without further qualification " geometry does not allow you to say that the sum of a triangle's angles is less than 180 degrees".
The sentence concerning the sum of the angles of a triangle is false in euclidian geometry, but not in geometry in general.
Is the situation analogous in logic? Would it be outdated to say, to someone that contradicts himself/ herself : " Logic does not allow you to say this".
To continue the comparison, I think that euclidian geometry is the geometry that is the most comformable to our everyday experience of the physical world, the most "convenient" geometry for ordinary purposes, and in that sense, a sentence that does not agree with euclidian geometry can be said "false" ( where "false" means : not corresponding to the facts of the world such as we ordinarily experience it).
Can classical logic be considered as the remaining standard in the same sense?
I think the situation is fairly analogous between "logic" and "geometry", but your assumption about what the situation is for "geometry" is wrong.
It is entirely common and non-controversial that one can say "geometry" without any further qualification when one is implicitly speaking about Euclidean plane or solid geometry only. If one wants to speak about other geometries, it is generally expected to warn the reader/listener of this first.
This is not any deep claim about what really "is", simply a convenient convention about the use of language.
Similarly, when you say "logic" without further qualification, it will usually be assumed that you mean classical first-order logic or classical propositional logic (which is effectively a subsystem) -- or possibly the kind of usual quasi-formal mathematical reasoning that classical first-order logic aims to encode.
The fact that we know many other specialized kinds of logic has not displaced what the word usually refers to.