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.
I like the idea of treating limewash as maintenance rather than a permanent cure. That is probably closer to how these materials were historically used anyway
I just restored an 1800’s house and we redid the paint in limewash as part of that process. (Effectively zero VOCs but pretty caustic when wet.)
Historically (for interior paint) you would use regular limewash in dry spaces and add casein/milk in high traffic/wet spaces. It works in a bathroom but wouldn’t hold up in a shower (for that you would want to use “tadelakt“ - especially if you’re going for that Aman spa look).
The contemporary solution with modern limewashing is to use “mineral shield” - it uses silicone instead of casein - it makes water droplets bead up but still lets water vapor breath through.
It also doesn’t flatten out the nice velvet texture one gets with limewash paint from the calcite crystals.
That said, it’s extremely obvious if one touches up a patch of wall with limewash. No two batches are the same color unless you’re going pure white and even then…
Lime wash also discourages people from leaning against the wall, once they have seen the white marks on thier suit...