Caulk is the way to go when there is a change of plane as there are shifts in the structure that will crack regular grout. You should clean all that mess out of there with a utility knife, single edge razor blades, a #10 scalpel, whatever. Clean out with some vinegar to kill any mold residue, let dry and re caulk with quality silicone caulk.
The only time grout would/could be used here is if you use the epoxy grout. It is more expensive and harder to work with but will yield great results.
By the way, all wet surfaces will mold over time without regular cleaning.
You would grout a location like this by placing mixed grout on the floor and ramming it into the crack with a grout float... something like a thick rubber trowel....