Blood of the covenant is thicker than the water of the womb

Nah, this one is actually came much later. The earliest use of the phrase comes from the 13th century and reads "kin-blood is not spoiled by water", referring specifically to the water used to baptise. The idea being that one's outlook on life will still match that of one's parents, and not be drastically adjusted merely by being christened.

By the 17th century the phrase as we know it had formed, with blood referring to kin. Over the 18th and 19th centuries the meaning of "blood" expanded somewhat, to refer not just to familial relations but also members of one's clan or nation.

The phrase you used is first attested in the 1990s and 2000s. Its authors claimed it was the original form of the phrase, but without citations to back up the claim.

It's possible that these authors are conflating the English phrase—which goes back in English to the 13th century and which as a philosophy goes back to ancient Greece and Rome—with a similar notion in Arab culture. There, they have the notion that blood-brothers are in a closer covenant than milk-brothers. Milk-brothers referring to the mother's breast milk shared by biological brothers, and blood-brothers being those who have chosen to form a blood pact, apparently involving literally licking each other's blood.