Hoop stress and radial stress act perpendicular to the center axis of the cylinder while axial stress acts parallel to the cylinder axis. therefore, either of the first two stresses combined with the axial stress is a multi-axial stress.
Ironically the term "axis" has four alternative primary definitions, of which, one is the center line line of a rotation, circle or curve"; and, a second is "one of the reference lines of a coordinate system". An alternative term to "multi-axial" stresses would be "multi-axis" stresses; but, based upon the Webster's dictionary either should be correct.
PS Just for your general information, in either case, multi- with a hyphen is the correct spelling.