I'll respond with a quote from Admiral Marcus in Star Trek Into Darkness, set 2259:
"The Klingons have conquered and occupied two planets that we know of and fired on our ships half a dozen times. They are coming our way."
Which is entirely incompatible with Discovery's open Klingon/Federation war in 2256/7.
So while there's no way DSC is the same world as TOS, it may look more like the Kelvin movies but it doesn't fit into that world, either. It's it's own thing.
The Klingons in the Kelvin reality are more similar to those of Discovery than those of TOS or TMP and TNG!
JB