Do 'skilling' and 'teaching' have different meanings? Yes. In general there are no exact synonyms. If they're spelled differently or pronounced differently, there is some slightest difference in how they are used.
'Skilling' was unknown to me until this question. It sounds like a two-step neologism process, first a verbification of the noun 'skill' into a verb, then taking the present participle 'skilling' with the intention of it being used like 'training'.
So your question really might be: Do 'skilling' and 'training' have different meanings? To be more specific than 'there are no exact synonyms', you really can't replace 'training' with 'skilling' because of the connotations; most people would have the same reaction as you to this strange new word. In context people would understand it to mean something like 'training' but with the extra 'very rare, only used in very particular business situations'. That's enough to make them different.
You may still be concerned that it is not a 'word'. More than one person has used it (internet search, ODO). More than one person has used it more than once. So even if there may not be any authoritative source that pronounces 'This is a word', it's being used consistently, and really in the end that's enough, at least for those people who've used it.
This sounds very similar to the 'gifting' controversy. Many people cringe at the the many other people who use 'gift' as a verb: "I gifted them a book" meaning "I gave them a book as a gift".
Words come in and out of fashion. 'Very' came from Old French after the 1066 invasion (cognate with 'vrai'), it replaced a perfectly good word from Old English that has a vestige in the rare 'sore' (as in "I am sore wroth" = "I am very angry") which is cognate with German 'sehr' = 'very'). Sure that annoyed the crap out of some poor saxons when the house staff took on airs and spoke fancy like their Norman overlords. They got over it eventually, or at least it didn't bother their kids as much.