Building fantasy counterparts of Earth?

A common trope in fantasy is modeling the fantasy world's geography and cultures after those of Earth's history. While there is nothing wrong with this besides being uninspired, it is rather arbitrary that this planet specifically resembles Earth and this setup needlessly constrains possible stories. Tolkien's Middle Earth set the standard for fantasy settings (even grimdark settings like the Warhammer World and Planetos are not immune) and after the umpteenth time it has become really stale.

The Only Fantasy World Map... ©2011-2017 EotBeholder

I feel that there are two alternatives to this:

1. Set the story on our Earth in any historical period with fantasy elements added, such as making The Lord of the Rings and Conan the Barbarian into historical accounts. This is the sort of setup present in Medieval Player's Manual, Dark Albion, Kingdoms of Legend, Chronica: Age of Exploration, and probably a few other settings I failed to mention.
2. Build a generic fantasy world that approximates Earth, and then shake it up by moving the geography around and spicing up the cultures. The final product will draw inspiration from Earth without being a clone.

Both of these are surprisingly difficult and both offer incongruously similar rewards. Both options strike a different balance between the familiar and foreign. A fantastical Earth must both retain a version of our own history and institutions, at least starting from a specific point of divergence, as well as work magic and monsters into the world that have always been present (barring some apocalypse where magic returns after a long absence).

The setting Kingdoms of Legend notably removes all mentions of Christianity and Islam to avoid offending audiences in favor of a fairly generic neo-Roman syncretism while incongruously retaining the political structures that only existed because of the former. Political correctness is an admirable goal, but it should not come at the cost of rational world building. Both Christianity and Islam have stories of pagans with magical powers and both religions hold that faith is required because the existence of God is not empirically testable, and therefore building the setting around these two tenants is the best way to respect their beliefs (as well as, you know, portraying worshipers as people with hopes, dreams and flaws rather than fanatical caricatures). Although it would be hilarious if the one true god turned out to be the Daemon Sultan Azathoth.

The benefit of a world that doesn't closely resemble Earth is that it defies expectations. You could easily add otherwise impossible political situations like Shogun Tokugawa being unwillingly betrothed to Queen Cleopatra while Montezuma tries to curry favor with both. You could throw together an ethnically diverse cast without having to write long backstories explaining why they traveled so far from home. A fantasy world with magical physics and chaos mutations is not constrained by the solar melanin hypothesis of human skin color (which may be discredited too).