My point is, it is possible that a sentence is true in one model and false in another (e. g. the 5th euclidean axiom). So there exist theorems that are only true in some of an axiomatics' models. So can we use models to prove theorems?
Can analytical proofs for euclidean geometry using the cartesian system be considered valid, as the cartesian system is only a model for the axioms?
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If you take a model, without any specific assumption on this model, and prove the formula in this model, then you have prove it in all models. You therefore get a theorem.
It's like taking a generic element $x$ when you have to prove a formula starting with $\forall x$.
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Your concern is well founded! One certainly cannot prove a theorem in a model and expect it to be true for the axiom system. This applies also to the model of Euclidean geometry given by Cartesian coordinates (developed simultaneously by Pierre Fermat and Descartes). The project of providing suitable axiomatics for Euclidean geometry was a major undertaking that kept David Hilbert and others busy for many years. There are many books analyzing the resulting axiomatics, including those of Szmielew:
Szmielew, Wanda. From affine to Euclidean geometry. An axiomatic approach. Translated from the Polish and with a preface by Maria Moszyńska. D. Reidel Publishing Co., Dordrecht-Boston, Mass.; PWN—Polish Scientific Publishers, Warsaw, 1983. xiii+194 pp. ISBN: 90-277-1243-3
and Hartshorne:
Hartshorne, Robin. Geometry: Euclid and beyond. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2000. xii+526 pp. ISBN: 0-387-98650-2
There is recent work in this area by authors such as John T. Baldwin and others.
In general, as the existing answers say working in a single model $M$ of a theory $T$ is only enough to provide negative information (if $M\not\models \varphi$ then $T\not\vdash\varphi$). However, if $T$ happens to be complete - that is, any two models of $T$ have the same first-order theories - then we can work in a single model without issue. Of course proving completeness is in general nontrivial, and lots of theories of interest are known to not be complete, but Euclidean geometry (appropriately set up) happens to be complete; this is due to Tarski (it's essentially a consequence of the completeness of the real closed field axioms).