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I've had multiple ideas of which real life cultures/cities Tyrosh is informed by in my brain, and I think I've landed on medieval Venice or Genoa (a maritime/mercantile city state that relies on the slave trade for continued economic prosperity that enslaves people of many different ethnic and religious backgrounds), as well as some al-Andalus (specifically as it relates to the Prague slave trade). The targeting of pagans by medieval states parallels what exists in ASOIAF canon of free folk being targeted. There's also the whole thing of the old gods being viewed as an illegitimate religion. While it is based on these culturally in my mind, Tyrosh is also kind of Poland because the Blackfyre rebellions are based on the Jacobite rebellions, and James Stuart (Daemon Blackfyre) married Maria Clementina Sobieska (Rohanne of Tyrosh) for her dowry and to support his cause as pretender to the British (Westerosi) throne. IWith all that in mind, Tyrosh doesn't have an exact parallel but it is greatly inspired by the historical record.

I think it is very important to recognize that Tyrosh is a slave state. There is a lot of irony many of the "Free" Cities heavily rely on slavery as an economic and social system. While Tyroshi wealth apparently has its origins in sea snails (why not), their existence as a mercantile society relies in large part on the slave trade. This isn't dissimilar to fellow slave state Lys, which has a relationship with the Targaryens, especially considering the amount of Valyrian descent amongst Lysene nobility. Lys parallels medieval Florence in the real world (which was also known for poisons of course). It's very.... interesting then that being sent to Lys was considered a punishment for Aerion (who might have been his generation's Valyrian supremacist if he didn't also think he was a dragon). Also, here's a quote from The World of Ice and Fire that makes the decision to send Aerion to Lys perhaps less of a punishment...

Even the Targaryen kings and princes of old sometimes turned to Lys in search of wives and paramours, for their blood as for their beauty.

I don't think there's the same thing in Tyrosh of going with the purpose of finding a 'proper' wife, but this provides the context that Targaryens have historically been willing to marry Free City nobility. However, the Blackfyre Tyroshi marriage (of Rohanne of Tyrosh and Daemon Blackfyre) is politically and economically motivated rather than by blood or beauty like it might have been for a Lysene noblewoman. Also don't pay too much attention to ages of the involved characters because it WILL give you a headache.

Despite the fact that slavery has long been outlawed in Westeros, the impact of the slave trade still impacts it. That the Blackfyre rebellion(s)/faction chooses to align themselves with Tyrosh is especially interesting. It shows that they care more about gaining financial backing than they do about upholding the moral code set forth by the Faith of the Seven/the people of Westeros that they intend to rule. Tyrosh has its own form of religion, more closely aligned to the Roman inspired religion practiced in Valyria (and the ancestors of the Targaryens). They are essentially the Catholic Jacobites to the CoE Englishmen (which gives so much context to the Targs being viewed as closer to God/the Seven). However, it isn't as if the people of Westeros aren't also benefiting from slavery, as the fancy armor of the nobility is Tyroshi in origin:

Tyroshi with a forked blue beard was selling ornate helms, gorgeous fantastical things wrought in the shapes of birds and beasts and chased with gold and silver. (The Hedge Knight)