It is important to distinguish between axiomatisation and interpretation. The mathematical formalism is nailed down by Kolmogoroff's axioms, but this only puts slight
restrictions on the philosophical interpretation. All three major philosophical interpretations are consistent with Kolmogoroff's axioms.
Secondly, what Kolmogoroff did was not actually an axiomatisation, but a definition. He defined that any measure that satisfied certain conditions was to be called a "probability measure". And the concept of measure was already defined as a function satisfying certain conditions. Now, it is traditional to call such conditions "axioms" but they are not axioms like the axioms of Logic or Set Theory. Kolmogoroff did not add any real axioms to the usual foundations of mathematics, he simply added a definition.
Kolmogoroff was not a completely committed frequentist, and he criticises the usual presentations of the frequency interpretation in his popular contribution to Mathematics etc. of which a translation was reprinted by MIT Press. He felt the frequency interpreation was useful in praxis but was logically circular and so could not be considered logically valid. Littlewood published a similar criticism.
BTW, "formalism", and axiomatisations are formal, never includes "interpretation". Klein and Hilbert famously insisted on this. Hilbert joked that the axioms of geometry should make equal sense if interpreted in terms of "tables, chairs, and beermugs" instead of planes, lines, and points.
In the 60's Kolmogoroff proposed an "algorithmic complexity" interpretation of probability and randomness. Eventually his proposal was worked out by Finnish mathematicians and is also discussed by Prob. von Plato, of Helsinki Univ., a famous logician and computer scientist.
See, for a careful discussion of the physical interpretation of probability,
http://rev-inv-ope.univ-paris1.fr/fileadmin/rev-inv-ope/files/35214/pdf_35214-09.pdf an open source journal, hosted at the Sorbonne, and, but less relevant, see also Prof. Leo
Corry's several publications on Hilbert's Sixth Problem in general, e.g.,
Hmmm... I guess jstor is not open access, is it. Prof. Corry's contribution to the ICM can be found for free on the Internet somewhere or other.