The propulsion was classified during the Klingon War. USS Discovery was a secret weapon against the Klingons. Information about the ship was not made publicly available, at least on it’s propulsion. Security on the ship was heightened to the point that Section 31 served on-board as seen in the 3rd episode of the first season. Burnham also wasn’t told anything about the drive until Lorca finally clued her in, using Captains authority. Tilly slammed up the second she realized that Burnham wasn’t read in as well. It’s discussed throughout the season by the Admirals that Discovery is their ‘secret weapon’ and if the Klingons found out about the ship it would be devastating to the war effort. The whole bit with L’rell, Ash Tyler and Lorca was all just to discover information about Discovery.

After the war, the ships spore drive was shelved until a non-human interface was found. However the red signals immediately popped up and Pike took command of the Discovery/Spore Drive to find out more about the signals. At this point, information about the ship wasn’t disseminated throughout Starfleet. The ship wasn’t classified but the propulsion was and Starfleet, presumably, didn’t see the need to spread this information further. When the ship was destroyed, Pike and Spock said that Control/Section 31 destroyed Discovery in an attempt to gain access to the spore drive and the sphere data. To ensure that this event couldn’t recur (due to time travel), everyone must never speak of Discovery, the crew, or the spore drive again. When Discovery arrived at 32nd Century Starfleet HQ, Admiral Vance said that Discovery was destroyed nearly a thousand years ago and that there was no mention of the Spore Drive, showing that the secrecy of the ships drive was so tight that not even Starfleet themselves knew about it hundreds of years later.