Each art medium has their own defining characteristics for modernity that are specific to the theory, techniques, etc of that art form, but I think they are all more-or-less summed up by Ezra Pound's rallying cry, "Make it new." After centuries of codifying tradition, passing forms down with modest evolutions with each generation, modernism is concerned with shattering expectations and making every form and subject available to the medium. So in poetry, that is often associated with free verse (where any rhythm and sound repetition is given poetic voice) and openness of subject; in visual arts, this is often abstraction (where any shape and color can be used for a painting, not just those that accurately represent something in the real world); etc.
In the case of Beethoven's piano sonatas, I'd actually argue they're not quite modern. This isn't a slight against the works, as I actually think they're something more difficult to grapple than that. There are certainly proto-modern moments, like the variation in his final sonata where it suddenly turns into a jazz jig straight out of a saloon player piano or the tonically incomprehensible climax to the fugue in the "Hammerklavier," and his late string quartets are highly modernistic, but the final sonatas seem to still be in too strong a conversation with classic forms, structure, and tonality to be truly modern. It's maybe the closing statements on that conversation--words so final that it forced modernity from its successors (though it's worth pointing out there's still romanticism to work through before form and tonality gets completely blown up)--but it still the conclusion of classical aesthetics using its language.
To illustrate this point a bit glibly, if any of these sonatas were composed today, they would undoubtedly be called "neoclassical." This is maybe unfair because any piece called a "sonata" would be flirting with that label right off the bat, but that is maybe my point. His sonatas are not where he made his overtures to modernity because a sonata is ineluctably classical, just like a sonnet can't be considered anything but neoformal. Anything making such heavy use of the sonata form in multi-movement works with largely traditional tonalism (his breaking of tonal rules is still the exception, not the expectation; he still more often sounds like Mozart than he sounds like Stravinsky or, most certainly, Schoenburg) is bound to classicism.
And this isn't to say Beethoven did not write modern music in his latter years. It is worth noting that Beethoven went half a decade between composing his final sonata before he passed away, and to my knowledge he had plans for more at the time of his death. From the time he penned the first sonata in 1795, this is the longest spell with no sonata in his career. You could easily argue that it is not death that interrupted his composition of more sonatas, but that he was indeed done with the form. Not only were his last sonatas so final that it left his successors speechless, but it may have even forced himself to move on from the form. So again, none of this is to doubt the modernistic spirit of Beethoven. He went on to write, among other things, the Grobe Fuge string quartet, which Stravinsky himself proclaimed would be "contemporary forever."
To return to the question, the trademarks of modernity are found in Beethoven's other late works, where anything is possible tonally (such as the chromatic theme of the Grobe Fuge), structurally (such as the single-movement 16-minute fugue of several incongruous themes being contrasted against one another), etc. These sorts of modernistic innovations that serve as seminal works for generations to come are the common form of innovation you find in art history. The breaking points where an artist does something new are almost easier to comprehend. The late piano sonatas are something more interesting to me. It is difficult to find a corollary of such a final period being placed at the end of an era as what Beethoven did not just with the final five sonatas, but with the entire body of his 32 piano sonatas. Someone who so thoroughly mastered, questioned, and reconsidered a form as to make it redundant is a feat so singular and spectacular, that it takes extraordinary evidence to convince a skeptic that Beethoven pulled it off.