Aside, I'm not sure what "and/or" means, but I don't think this describes the actual behaviour:

julia> eachindex([1,2,3], [1 2 3])  # size (3,) and (1, 3) accepted
Base.OneTo(3)

julia> eachindex([1,2,3], [1 2 3]')  # size (3,) and (3,1) not accepted here
ERROR: DimensionMismatch: all inputs to eachindex must have the same axes

julia> eachindex([1,2,3], [1;2;3;;])  # size (3,) and (3,1) now accepted, as both prefer linear indexing?
Base.OneTo(3)

julia> eachindex([1;2;3;;], [1 2 3]')  # both size (3,1), but different eachindex
CartesianIndices((3, 1))

Is this what's intended? And how should it then be described?

In English "or" often means xor, so "and/or" disambiguates that case. It's a bit awkward.
https://en.wikipedia.org/wiki/And/or

Ok, although it still seems an odd choice to describe computer logic, not a menu. And if differing dimensionalities means ndims, that implies differing size anyway.

While differing size might be necessary for an error, it's not sufficient. Can we describe the actual behaviour accurately? Or are some of those bugs?