What does "monotonicity" mean in the context of mutability
The author appears to be using the same (general) concept of "monotonicity" as in pure mathematics.

Using the example of a vector, if the size of some particular instance of vector is monotonic, then it would seem reasonable to assume that the memory location of any previously pushed element will not be subsequently modified, and thus that it is safe to directly change the value of that element without having to synchronize with the vector's subsequent push operations.

This assumption seems reasonable for much the same reason why it seems reasonable to assume that the memory location of an element previously pushed onto a (non-circular, array-based, unbounded) stack whose size is strictly increasing will not be affected by a future push operation.  This stands in contrast to a stack on which both push and pop operations are being invoked, and so the memory location of a previously pushed element may be "removed" by a pop operation, and thus available to be assigned to once again (i.e. modified) by a subsequent push operation.

The author's point is that this assumption is not correct, since even in a data structure whose size is strictly increasing, future insertions may affect previously inserted elements by triggering an internal reallocation of some sort that is distinct from the "logical" effect of the insertion.