Everything that could connect Dottore to Shiruyeh/Kavusbay and the Jinn, 6.1 main spoilers included.

-His name Zandik: Zandik was a degratory term meaning "Heretic" used for those who opposed the Zoroastrian Orthodoxy in the Sassanid Empire, it still has the same negative connotations Islamically.

It's most likely that this Primal Obelisk:

...These followers were questioned and admonished by the orthodox priests of the mistress of flowers, and those found heterodox were... punishment...

references the Zoroastrian Orthodox Priesthood, however Zandik might not be his real name and considering the huge Zoroastrian influences on the Akademiya they probably would have been the ones to have come up with the term.

It probably also has to do with the most likely major inspiration behind his character being the Persian physician Abu Bakr al-Razi, more information on all that in this Sumeru lore post.

Genshin wise, it's pretty easy to see connection with "Shiruyeh" being also a degratory term in the desert's culture, him being associated with his "Brass Mask", but those are all pretty loose.

-Dottore and plague doctors.

Obviously there's Shiruyeh's plague which is based on the actual plague of Sheroe.

Dottore's constellation represents a 17th century plague doctors mask.

Now it is pretty odd why they would choose plague doctors a European phenomenon for a Persian inspired character, it might have to do with the Italian influences on the Fatui as a whole and I have my reasons to believe that they could try to connect those two completely unrelated irl events lore wise.

-Raven symbolism: If it wasn't obvious the bird depicted on his outfit is a raven which is pretty interesting, especially due to Liloupar's descendant ways of addressing the Shepherd boy throughout the books The Shepherd and the Magic Bottle and The Tale of Shiruyeh and Shirin:

"The ravens of the mortal world are so exceptionally noisy..." A Jinni arose from within the bottle in the form of a dancer, as flexible and languid as a cat and as bright and beautiful as a water lily. "Ravens...?" The youth was clueless, as he did not resemble a bird, nor was he noisy. "Yes, ravens." The spirit repeated, rather impatiently. "Birds that live fast and die fast, a noisy lot and no mistake... 'Only when they die do they wake up.' What about this is there to not understand, hmm?"

-Dottore's Segments: A Segment is something that is divided from for example an apple, when you slice it you get it's segments.

This word implies that Dottore's Segments were originally part of him, Liloupar's punishment:

.. Split... the Jinn-Mother... divine spirit... parts... ... The seven sages... hid them separately... ...To her was given infinite time... the hope of salvation...

-His shapeshifting abilities: this one is rather simple, a common trait of the Jinn like many other local deities (Yokai, Adepti etc..) and interestingly they can also do this:

Jinn are believed to be capable of possessing a human's body, which one Jinni uses as a threat against the human boy she dubs "Raven."

His personality overall is pretty similar to two specific jinn we know lore wise- Liloupar and Makhaira.

🛑 6.1 event Spoilers.

!-He's not human?!<

!It's already known he isn't a "normal" human; but it's never explained why he has such a long life span, certain powers (including the shapeshifting) and all that, for now it's only guesses.!<

!In the event the Ibis King addresses Makhaira a jinni as a human, now this isn't the first time this happen in game as seen with the Traveler as recently as Rerir but even if the Ibis King probably did not look that human in his prime he still must have known that Makhaira wasn't one either, on top of that we finally hear about Zandik again in the same event she is mentioned.!<

!Compared to Goudarz's deal with Thoth hers seems really unnecessary to the lore we already knew about her till now so it's interesting that they added this.!<

!Alternatively he could also just be a human who was influenced by a jinni and gifted those still unexplained powers, lots of possibilities.!<

Anyway this is not really a "theory", more like potential connections, hope you enjoyed it and in case I missed further possible connections do tell me, Hoyo really need to go deeper into his past imo.