GEM website has some good information about koji. As do the
books by sandor katzz.
Unlike a, SCOBY, yeast, or bacteria starter like you mentioned, koji is a mould and seems to taste best if it can go through the life cycle rather than a continuous starter. It can be kept dormant at the spore stage.
It's pretty easy to work with, but can get sour if it doesn't have the right temperature or humidity. I'm not a fan of our long ferment miso but my Japanese friends love it as it tastes like expensive artisan-made miso. Shorter ferments of under 3 months are more my style.
These days I mostly buy the koji rice instead of inoculating it myself as it works out cheaper. But it's sake koji rice so really best for shiro koji and amazuki.
Sorry, this spellchecker doesn't work with japanese.