Since the two phrases you quote appeared as part of a comedy sketch, I think it makes sense to consider how and in what sense the use of the word bitch may be played for laughs.
First, though, I note that the frequency of the term "bitch" in Google Books search results has increased considerably in the past 100 years—and especially in the past 50 years—even after we factor out the rise in frequency of "son of a bitch." The following Ngram chart shows the relatively frequency of "bitch" (blue line) and of "son of a bitch" (red line):
The fairly steady level of usage for bitch between 1800 and 1920 reflects the fact that in most matches for the term during that period involved the literal "female dog" meaning of the term. I don't know whether usage of the word in that original sense has increased or decreased since 1920, but it seems unlikely to account for much of the rise in the blue line since 1920.
Likewise, though the frequency of "son of a bitch" has increased significantly from its near-zero level at the beginning of the 1920s, removing the percentage of the blue line attributable to it leaves a major rise in the frequency of occurrence of bitch still unaccounted for. The remaining increase in the blue line's frequency, I suspect, consists primarily of instances where bitch is used as an insulting term for female human beings, although there are undoubtedly also instances where the word is attached work, karma, and other nonhuman objects.
Use of bitch as an insult term in published writing appears to have grown immensely in recent decades, and this development reflects an increased level of familiarity with and acceptance of the term in everyday English speech. I daresay that many (but not all) other derogatory terms and slurs have also grown in frequency as writers seek to mimic the natural language patterns and vocabulary of people on the street.
But the use of bitch in a TV news skit by a smart and widely admired actor and writer (Tina Fey) seems to indicate that something is going on other than an uncritical embracing of the word. I suspect that the skit is (or was when it appeared in 2008) in part a comedy-edged attempt at reappropriation, which Wikipedia defines as "the cultural process by which a group reclaims—re-appropriates—terms or artifacts that were previously used in a way disparaging of that group."
Instead of fleeing from or decrying the term bitch when critics applied it to Hillary Clinton, Fey in her news skit turned the usage back on its users, suggesting that hostile people resort to the insult to try to discredit effective women—women who "get things done."
The final comment in the skit, "Bitch is the new black," if taken literally, would amount to saying "Bitch is the new height of fashion," as the phrase "X is the new black" does when it refers to the highlighted color of a new season's lines of clothing. But that's not what Fey's comment—absurd though it appears on its face—seems to be aiming at. Again her main point seems to be an assertion that women are becoming more willing to expose themselves to the insult of being called a bitch because they understand that hostile use of the epithet often occurs in reaction to their prominence and effectiveness.
The two quoted sentences appeared in a comedic setting, but the aim, I think, was to reappropriate—or at least to nullify the injuriousness of—the insult bitch by redefining what it means. Though I haven't read the column by Maureen Dowd, I suspect that she brought up the Tina Fey skit to make a similar point about the need for women (and Hillary Clinton in particular) to admit to being tough sometimes instead of trying to persuade the public that they are actually sweet and benevolent to the core.
