Generally you want tough, resilient wood that isn't prone to splintering. The list of woods that meet all three requirements to a certain degree is looong. That there's a much shorter list of woods that have been and/or are currently favoured for walking sticks might seem to show that if you can pick and choose you select stuff that's particularly strong but actually a great deal of what was chosen historically came down to its looks (colour as well as grain/structure) and not due to its inherent great strength.
A sizeable chunk of walking stick production uses raw material where the stick in its entirety is supplied by the raw material, i.e. bamboo shafts, branches or saplings, which then require minimal processing (rather than forming a dowel/spindle by cutting away material from a board or riven length). While this can be inherently strong material with no weak axis it shouldn't be forgotten that their stock would be essentially free. Any wonder a craft of this sort would grow to favour a raw material that didn't cost them anything! So the species that are chosen for this may not be as superbly well suited as their glowing marketing says they are.
Beyond what certain species can provide you do have to select your stock carefully because all strong woods can have weak individual examples. So just as was said in an Answer from a long time back you should ignore the species, and ask yourself "What's this piece like?" That was in relation to tool handles (for things like axes) where the requirements are much more stringent but it can still apply here to a lesser degree.