Simply put does the capstone of the Stoneblessed class from Races of Stone grant access to the Racial Paragon Class from Unearthed Arcana for the respective race?
IE does a Stoneblessed(Dwarf) 3 have access to the Dwarf Paragon
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for all effects related to race, a stoneblessed is considered a member of the race to which she is bonded. [...] The stoneblessed meets any racial prerequisite for prestige classes and feats as if she were a member of her bonded race.
(Description of the stoneborn ability of the stoneblessed prestige class, Races of Stone pg. 123)
So there are basically two ways in which a stoneblessed might be able to qualify for the racial paragon class of her bonded race: if qualifying were an effect, or if it meant meeting the prerequisite of a feat or prestige class.
Unclear—the term “effect” isn’t really well-defined in general. Spells (and similar things) have a fairly well-defined “effect” entry, but that doesn’t even cover the entirety of a “spell effect,” since the latter would be the sum total of what a spell does, not just its “effect” entry (which many spells don’t even have). Outside spells and other similar things with such well-defined statblocks, we don’t even have that much to go on.
This question has come up before, and then as now, the answer has to be “we don’t really know.” I would hazard that for most purposes—including stoneborn—it’s safe to consider a prerequisite an “effect” in the context of rules saying you count as X for effects. It’s possible that there are some weird corner-cases that would require more adjudication, or even just nixing the combo, but I can’t think of any and would feel comfortable handling that on an ad hoc basis.
But nonetheless, RAW here is rather murky.
This one is rather clearer—the answer is “no.” Feats are obviously not remotely similar, and as for prestige classes, Unearthed Arcana never refers to racial paragon classes as such, and in fact lists them alongside base (“standard”) and prestige classes as a third category:
A character can multiclass freely between standard character classes, prestige classes for which he or she qualifies, and the character's appropriate racial paragon class.
(Racial Paragon Classes description, Unearthed Arcana pg. 32)
So RAW, the bit about feats and prestige classes doesn’t cover racial paragon classes.
RAW, you’re on shaky ground here.
However, Unearthed Arcana is a book of optional variant rules. None of them are officially incorporated into the game. As such, Races of Stone was never going to mention or acknowledge them—as far as Races of Stone is concerned, they don’t exist. Therefore, there was no way that stoneborn was going to include them in the list alongside the prerequisites of feats and prestige classes. Stoneborn is supposed to make you simply count as your bonded race, no if’s, and’s, or but’s. And racial paragon classes are effectively mini-prestige classes, for all Unearthed Arcana puts them in another category. Hell, Unearthed Arcana even emphasizes the “prestige” of being a racial paragon. Racial paragon classes aren’t an intentional omission; they’re an obscure technically-distinct concept that are fundamentally exactly the kind of thing that stoneborn was written to cover.
Furthermore, as I suggested in the linked answer about “effects,” most of the time, it’s best to consider “whether or not you can take levels in a class” as an effect. You count as your bonded race, so that should include the prerequisite checking you. Even if they left off the mention of feats and prestige classes, I would include them for stoneblessed, and I’d include racial paragon classes, too. To me, anyway, if they were supposed to be left out, the book should have called that out explicitly.
Finally, remember what Unearthed Arcana is: a book of variant ideas that a DM can take and run with if they like. Most of those ideas aren’t fully fleshed out. They certainly aren’t incorporated into the wider ruleset and handled by other rules. Any time a DM uses them, it becomes the DM’s responsibility to flesh out the details—like how they interact with all the rules elements from other sources that have been written assuming they didn’t exist. Ultimately, this is just another case of that, and that’s what we should always expect with Unearthed Arcana.