There are a couple of pretty good grout colored caulks out there. But if you are careful, the grout should not have enough water to discolor or warp the wood floor. I would try that first.

Your only choice right now is to use a grout caulk. Grout or mortar will not stay there without cracking/crumbling out.

A small gap between the wood and stone is what is recommended...that gets carefully filled with an elastic caulk.

There is usually some disagreement amongst installers as to how much expansion space is required for a wood floor to perform well and not to have problems due to being locked in place. I have seen solid wood floor cut net to vertical surfaces and they never had a problem. However, I can definitely think of houses where cutting net to vertical surfaces would be risky, if not downright foolish.

Flexible caulk of a close match is what I'd use.

I usually undercut the stone and slide the flooring underneath, but now you have a gap to be caulked.

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