The way to settle this is to remember that the medial and lateral menisci behave oppositely because of how they are tied down.
The medial meniscus is locked to the joint by its firm attachment to the medial collateral ligament and the capsule. A structure that is held tight cannot move much, so it is the less mobile of the two. That immediately marks the claim that the medial meniscus is more mobile as untrue.
The flip side of being fixed is that it cannot dodge sudden twisting loads, so it tears more often. That confirms the medial meniscus is more commonly injured. The lateral meniscus, being almost a full circle and loosely held, slides freely and also drapes over a greater share of its side of the tibial plateau, so the coverage statement is correct. Finally, like all fibrocartilage, menisci are built mostly from type I collagen, so that statement is correct as well.
Only the mobility claim breaks the pattern, so it is the false one.
\[\boxed{\text{Medial meniscus is more mobile than lateral}}\]