I'm not sure as I'm not a builder or plasterer and it does depend on the use case.

However I do know that some grout and some paints that say they are anti-mold contain Lime or calcium hydroxide as their primary anti-mold ingredient. I think pure Lime putty can be used as pointing material between bricks in humid basements. Lime plaster (lime mixed with sand) can also be used in bathrooms (with some considerations I am not familiar with). So, my best guess is that some mix of Lime can be used as grout. But I can't say for sure.

Something I plan on doing before this winter is to use Lime Wash. Take Lime (CaOH2 in powder form) and mix with water until it has milk like consistency and brush it on grout or on (white) bathroom walls. If it works, I expect it to be a yearly or quarterly ritual and not a one time solution.

EDIT: A fascinating historical material I recently learned about used in bathrooms is called Tadelakt [1] which uses Lime as one of its ingredients.

[1] https://en.wikipedia.org/wiki/Tadelakt

In my experience, grout doesn't go mouldy but silicone sealant certainly does. It's the caulk and the sealant that you want, i think, to replace

There is a variation of lime wash that includes a fatty oil as a sealant and has been around for 100s of years

https://en.wikipedia.org/wiki/Tadelakt

In India, people have used lime for all sorts of things in houses, including grout in places. It requires yearly maintenance.

Pure silicone caulk is mold-proof, even in the white variation. The various "improvements" in many caulks allow mold to get a foothold.

it's not waterproof