Learn Farsi: شباهت بین این ژن‌ها اینکه این خانواده اصالت دارد را ثابت می‌کند. - The similarity between these genes proves the fact that this family has authenticity.

In this sentence, the entire clause اینکه این خانواده اصالت دارد (the fact that this family has authenticity) acts as a single, large noun. Because this whole phrase is the direct object being proved, the object marker را goes at the very end of the clause.

Could we just use که instead of اینکه to mean "that"?

اینکه is much better here because the clause is acting as a grammatical object. Adding این (this) gives the abstract idea a concrete "thing" for the sentence to grab onto (literally: proves this, that the family is authentic), which makes it much more natural to attach the object marker را at the end.

Does بین (between) need an ezafe connector here?

Yes, it is pronounced beyn-e in zhen-hā. Like most Farsi prepositions of place or position, بین requires the short "e" sound (ezafe) to connect it to the noun that follows.

Is اصالت دارد a compound verb?

Not exactly; it's a noun plus a simple verb, literally meaning "has authenticity" (اصالت means authenticity, originality, or noble roots). Where English often uses "to be" with an adjective (is authentic), Farsi frequently uses "to have" with an abstract noun to express a trait.