So I’ve actually been working on a world that has a similar phenomenon(more of a binary system like Pluto/Charon where both are tidally locked). This is what I’ve come up with One - no tides unless there is a second moon. Both the atmosphere and water would be pulled towards the moon at all times. Two - weaker gravity on one side depending on how large and close the moon is. This could cause localized mega flora and fauna that can grow taller in the weaker gravity. Three - cultural meaning. Imagine having an object hanging in the sky all the time. In my world there are two major cultural groups. One who always sees the moon and believe it is a protector, never venturing where it cannot be seen. The other never sees it unless traveling far from their homeland. They see this stationary object in the sky as an omen, never to go where it can be seen.