Both Scottish English and Scots descend from the same root Anglic, but there's been divergence since then -- in the same way Scottish Gaelic and Irish Gaelic are descended from the same root, are somewhat mutually intelligible, but are treated as two different languages. It's useful to make that distinction simply because there is a degree of unintelligibility.
Everyone on this website is speaking an Indo-European language, and I'd bet a majority of them have an Indo-European language as their first language. They're all from the same root, so are they all speaking dialects of Indo-European? Or are they speaking distinct languages? Where, exactly, you decide "this was a dialect, but now it's a language!" I can't say, but that point surely exists. It's obviously muddied by politics and national identity and a hundred other factors, but at some point, surely, a language ceases to be a dialect and becomes a language of its own.
I have no real authority, but to me, I see no reason why Scots shouldn't be a language. English and Scots would certainly not be the most similar languages to be considered distinct. Plenty of other varieties of the English continuum, for that matter.
Categorization is easy when a difference is so large as to be irrefutable. But thinking in terms of continuum is much more useful when looking at small differences.
Language is a social construct. We're colloquial beings, and 99% of the time we all agree on what's what. What irritates me are the folks that think that the remaining 1% can be decided in some kind of objective way. Is Scots its own language? Depends on who you ask, and in what context. There are valid arguments either way. Embrace the ambiguity.