If it wuz me, I'd probably just use a sanded floor tile grout that matches (as closely as practical) the colour of the tiles or the existing floor tile grout?

Just paint the surfaces you want the grout to stick to with white wood glue and allow to dry, and mix some white wood glue into the grout when you're mixing it. The addition of the glue will make the grout stick to the wood better, and give it more flexibility and elasticity so that it can better accomodate any movement in the wood.

Ideally here, you'd want to use a concrete bonding agent instead of white wood glue. White wood glue will re-emusify if it gets wet and stays wet for a while whereas a concrete bonding agent will have a time window after it dries where it will still re-emulsify due to the moisture of the wet grout, but after that time window has elapsed, it will be unaffected by moisture. However, it's much more probable that you have white wood glue on hand, and equally probable that you'd have to pay $50 for a gallon of concrete bonding agent. So, if you use an acrylic grout sealer to paint over the grout after it sets, you could keep the glue dry and get away with using white wood glue instead of a concrete bonding agent like Daraweld C by the Grace company.