Why Michael, vsause's host at the start of each video says "Michael here" instead of "Michael is here"?
Is it grammatically correct or just a slang / simplified way of speaking?
It's not an abbreviation of 'Michael is here'; rather of 'This is Michael here': compare the BBC wartime introduction 'Ici Londres'. This is a not uncommon way of introducing yourself, though it always sounds a little strange to me.
And 'grammatically correct' in your question also sounds slightly strange. No, this is not a complete sentence, but there is no reason why it should be; headlines, interjections, introductions and the like need to be understood rather than to be precise.