Usage of double dots (..) Is it formal?

As far as I know, double dots isn't actually an established punctuation mark (unlike the three-dot ellipsis), but you can see it a lot in informal written online communications.

I think that it comes from overuse of the ellipsis. The ellipsis is overused in emails/Facebook statuses/chats because it's a rather vague and unspecific mark when people sort of ramble with their thoughts and don't frame them in well-defined sentences. I'm not saying this critically or pejoratively but as an observation on the less-formally structured form of online communications.

So the double-dot, I believe, represents a grass-roots construction that's like a less-dramatic and less-drastic ellipsis. It is perhaps intended to be a shorter pause -- less meandering and long-winded.

A double dot is also used in mathematics for integer intervals (e.g., [−1..2] denotes the set {−1, 0, 1, 2}). It is also used in several programming languages to for similar concepts.

A legitimate use of dots is in a quotation from which a part is missing. This piece of punctuation is, as others have said, ellipsis and normally consists of three dots. The only place I have seen two dots is in quotations in the Oxford English Dictionary, where they are used to save space.

As a young copywriter, I first noticed use of the two-dot ellipsis as a space saver in Sears Roebuck's 1964 catalogs.