Discovery is effectively a reboot with the numbers filed off.
So were TMP and TWOK, really (there were fans at the time who insisted they couldn't
possibly be in the same reality as TOS because of all the changes), but people have had decades to get used to the differences and rationalize them in their minds. And so was TNG to start with; Roddenberry wanted to keep as far away as possible from TOS ideas and redo Trek in a new and improved way, but then he was succeeded by writers who were TOS fans and who chose to tie the shows together more directly.
The thing that needs to be understood about continuity is that it's not the sole, overriding parameter for analyzing a work of fiction; it's just one more story device that storytellers make up along with characters, plots, settings, and the rest. So who's telling the story ultimately matters more than what universe it pretends to be in.
Any new version of a fictional premise by new creators is going to be a reinterpretation and a different take on things to some degree, whether it pretends to be in the same continuity or not.
But most everything from aesthetics to character developments to the look of the Klingons could, rationally, have been precursors to the Kelvinverse, but not so easily to TOS and the others.
Not precursors, since they're more or less simultaneous.
Discovery season 1 takes place in 2256-7, which is during the first Kelvin movie, specifically in the gap between Kirk joining the Academy (2255) and the main body of the film (2258). Which is exactly why they can't be in the same continuity -- DSC has the
Enterprise already in service a year or two before it's launched in Kelvin, and DSC has a Klingon war in 2256-7 when
Into Darkness in 2259 portrays a Klingon war as something that Admiral Marcus thinks is likely but that clearly hasn't happened yet.
The aesthetics are similar because they hired some of the same designers and are making it for the tastes of the same generation of viewers. That's all. It doesn't make sense to base perceptions of continuity on artistic style. John Byrne's Superman comics have the same aesthetic as his X-Men comics, but they're certainly not in the same universe.