Why did Serre choose coherent sheaves?

All Serre needed to use was quasicoherent sheaves. For these it's a fundamental theorem that Cech cohomology is the "correct" cohomology. That means that it agrees with the cohomology defined abstractly, via derived functors. Since derived functors give rise to long exact sequences, the fundamental theorem implies that short exact sequences of (quasi)coherent sheaves yield long exact sequences in Cech cohomology. This just isn't true for coarser abelian categories like arbitrary sheaves or presheaves.

For an idea of why the theorem is true for quasicoherent sheaves, it's generally true that Cech cohomology becomes the right cohomology when the space has a "good cover," which in this case means a cover whose elements have no higher cohomology for whatever sheaf we're investigating. This is always true for quasicoherent sheaves by a reduction to affine varieties and then commutative algebra, which isn't possible for general sheaves since there may be no cover on which the sheaf is that associated to a module over a ring.

Serre didn't mention quasi-coherent sheaves nor derived functor cohomology (=Grothendieck Tohoku cohomology, introduced in 1957) when he finished writing FAC in October 1954, because nobody in the world knew about these at that date.
He didn't know about abelian categories either for the same reason.
He introduced coherent sheaves in algebraic geometry in imitation of the notion of coherent sheaves in analytic geometry, where they had been created by Oka and formalized by Henri Cartan and Serre himself.
In particular one of the triumphs of FAC was the proof in the algebraic geometry context of the Cartan-Serre theorems A and B.
The only sheaf cohomology used by Serre was Čech cohomology.
The demonstration that one could use the absurdly coarse Zariski topology to obtain powerful results for algebraic varieties stunned algebraic geometers, foremost among them Zariski (!) and Grothendieck, who was initiated to algebraic geometry through FAC.