those ships are only partly made of organic tissue (mainly the hull/armor as well as some interior interfaces) without the ship itself being a living organism
when speaking of an organic ship i've got for example lexx in mind, which is an actual (yet artificially created) huge living organism (e.g. like the drakes or voidspawns in stellaris)
The Lexx was created as a flagship for His Divine Shadow to use on a universe-wide cleansing. It is the spiritual successor to the previous flagship the Megashadow (and its own predecessor the Foreshadow). “I am The Lexx, I am the most powerful weapon of destruction in the two universes” — The...
lexx.fandom.com
this kind of ships are usually part of a hive mind or are controlled by a psionic entity, there are alao cases where the ship is fully self-aware and acts independently just like any other living organism
lexx for example was built by the divine order which itself was an insectoid hive fighting off humanoids and is fully self-aware, the new crew made up of individuals it treats more like kids to protect than