Have to nitpick first - Microsoft isn't an application, it's a company that produces software that includes operating systems as well as applications
Just about every application designed for an international audience has both versions and will show either or both as correct depending on your location settings.
My computer shows only 'axe' as correct, whereas my phone recognises both 'ax' and 'axe'.
If the popularisation of 'ax' is because of computers, it's solely down to lazy users leaving everything as default - but I think it's more to do with the general trend toward Americanisation that's seeping into everything (some of it is good, some bad - I'm not getting into which bits I think fall on which side
)