Is the SE "new privilege" notification message a sentence?

It certainly is not a grammatically correct sentence, but at its core it has a valid sentence structure, "you've gained the privilege". It would approach correctness rather more with to in place of the dash; a period after answers; and learn capitalized. There may be some reason (aside from a mock-casual tone) for abbreviating Congratulations; if there's no such reason, it might as well be spelled out.

I don't think that it's perfectly punctuated.

Congrats, you've gained the privilege – edit questions and answers

Setting aside the learn more, there is no period at the end of the sentence. But, I'm guessing that that's not the issue here. It's the dashing dash which is, and IMHO, it's being used incorrectly. For starters, it's an en dash (assuming that it's been pasted correctly), and it's not really kosher to use it to replace a colon (which is what is being done here). This dash is essentially used only for one thing, indicating ranges such as 1–10, 1900–2000, and so on. The em dash, on the other hand, is sometimes used in informal writing to replace the colon.

A hyphen, an en dash and an em dash: - – —.

The above should illustrate which dash is being used in Atwood's sentence. This line can be fixed to use the em dash (sans the spaces on either side) as below:

Congrats, you've gained the privilege—edit questions and answers.

The differences between hyphens, dashes, and the minus sign, are outlined here.

Grammatically correct, the sentence is.

A similar sentence structure is:

You've gained the sword: "Sting".

As wikipedia states:

Em dashes are sometimes used to set off summaries or definitions

Since "Congrats, you've gained the privilege." is grammatically correct as sentences go, adding a definition of the privilege is also fine.

Learn more is a link and so is not part of the sentence. It might be better to separate it with a full stop.