Fuck no, Discovery is not the Prime Universe, regardless what anyone says. Honestly, at this point I don't get why they insist on romanticizing a continuity that hasn't been active since 2005, but then even when Star Trek admitted they were doing a different timeline they still kept it as a branch off of the Prime Universe as a means of keeping Leonard Nimoy around.
But really, wouldn't it be better for all concerned if Discovery were a separate continuity? Then they could have some freedom to move the storyline wherever they wanted. As it is, we know the Klingon war burns itself out and spore drive turns out to be a dud if we adhere to the Party Line that this is the Prime Universe.
People lose their minds over the "D7" Klingon ship designation in DSC. Where in canon sources was the configuration we see in TOS referred to as a D7? A DS9 tribute episode 30 years later? Cmon kids there's more important shit to lose your mind over.
Actually,. that's not really as silly as it sounds. The simple truth of the matter is, in order to know what that Klingons have a ship design known as a D-7, you'd have to look it up. Which means the writers probably checked out the Encyclopedia, Memory Alpha, or they just Googled it. All three options include images of the D-7. So by labeling a different design "D-7" they actively chose to ignore the very canon they claim they are upholding.
And really, the whole debacle was easily avoidable, they could have either have used a different name, or even just used the original D-7 design. And don't anyone pull the "it was designed in the 60s, it has no place on modern TV" line. Freakin Abrams kept the D-7 design mostly unchanged in Trek XI. If the design is good enough for a big budget theatrical movie in 2009, it's good enough for a TV show in 2017.
Is there a reason hair should be mentioned?
Well, for Klingons, their hair is part of their mythology. Kahless made the first bat'leth by shaping a strand of his hair and then dipping it into lava and hardening it into that shape. Kind of hard for someone who belongs to a hairless race to pull that off, even if he is their messiah.