Kolmogorov's axiomatization achieved greater simplicity, clarity, rigor and unification of applications by grounding probability in measure theory. There were prior axiomatizations that went in a different direction, e.g. Bernstein's algebraic approach (1917), or von Mises's frequentist one, see Lambalgen, Von Mises' Axiomatisation of Random Sequences. Both are sometimes named "the first" axiomatizations of probability.
However, Bohlmann got there much earlier, sort of, see Bingham, Measure into probability: From Lebesgue to Kolmogorov. Hilbert refers to his labeling of the rules of total and compound probability as "axioms" in a footnote to the sixth problem from his famous list of 1900:
"To treat... by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theories of Probability and Mechanics".
Bohlmann's work was published in the Encyclopadia der Mathematischen Wissenschaften in 1901, and later in the proceedings of the 1908 International Congress of Mathematicians. He gave the first rigorous definition of independence, and distinguished between independence and pairwise independence, but did not give what one would call an axiomatic development, i.e. derived any major theorems. His work was, apparently, unknown to Kolmogorov.
The developments that influenced Kolmogorov's measure-theoretic axiomatization are described by Barone-Novikoff in A history of the axiomatic formulation of probability from Borel to Kolmogorov, see also The origins and legacy of Kolmogorov's Grundbegriffe. For a summary of Borel's contributions and ill-advised choices that hampered his work see Why did Borel reject countable additivity of probability? Sierpinski's axiomatization of measure theory (1918) adapted to Borel's probability framework by Steinhaus (1923) were also major stepping stones. From Barone-Novikoff's introduction:
"The need for an axiomatic foundation for Probability Theory had been stressed by Hilbert as part of the sixth problem in his celebrated list of 1900... The landmark paper, initiating the modern theory of probability, is E. Borel's "Les Probabilities Dénombrables et Leurs Applications Arithmétiques" of 1909... The key figures immediately following Borel are G. Faber, F. Bernstein, and F. Hausdorff.... The ironical circumstance [is] that Borel, the unquestioned founder of measure theory, attempted in 1909 to found a new theory of "denumerable probability" without relying on measure theory. The irony is further compounded in the light of Borel's paper of 1905 which identified "continuous probability" in the unit interval with measure theory.
[...] Steinhaus (1923) finally incorporated Borel's "general theory" into measure theory by exploiting an axiomatic characterization of measure theory due to Sierpinski (1918). Steinhaus' work and more generally the increasing abstraction of measure theory itself (initiated by Frechet (1915) and Caratheodory (1914)) were events which helped pave the way to Kolmogorov's achievement."