How do I repair the grout on my bathroom floor?

You can scuff the old grout and add new grout. As far as matching, even if you use the same grout it will be off due to aging. You can paint the grout for the room and then seal it. I used outdoor paint and an artist's brush when I did my bathroom floor.

It may be simpler to add quarter round to the baseboards.

I agree, it is almost impossible to match grout colors. I would either paint/stain the grout (available at most home improvement stores) or if you want to do the work, grind out the old and re-grout. It doesn't look bad from the pic and the grout should be lower than the tile so if you don't want the gap, a quarter-round edging would solve the problem.

Perhaps you'll find something here that'll work for you: https://www.youtube.com/results?search_query=how+to+repair+grout+on+tile+floor

You can get caulking in a variety of colors, one close enough to whatever grout color you have, to use around the bottom of the base boards. This also helps with any water from the bathroom running under the wooden baseboard and causing long term damage and mold issues. Another, remedy would be to remove all of the wood baseboards and replace with the rubberized base molding. This is a relatively cheap and easy to install product. It has a little curve to the bottom edge that would over lab the rough grout of the outer tile edge.

If the bathroom isn't too large you can scrape the old grout and re-grout. Take about 1/4" of grout out to get a clean surface for new grout to stick to, clean with a vacumn, pick out a new grout color and do it.

https://www.homedepot.com/p/QEP-Professional-Handheld-Grout-Saw-for-Cleaning-Stripping-and-Removing-Grout-10092Q/203365414

To get the grout out.

As an old homeowner who is dealing with bathroom floor problems because of poorly done repairs--PLEASE re-grout or caulk, THEN use the quarter round moulding, which is great for hiding a multitude of sins.

My personal opinion, the rubberized moulding always looked cheap to me, besides reminding me of hospitals & other institutional usage. Every rental I was in that had it, the stuff didn't want to stay on the wall.