John in NM
At a very rough guess, the whiter parts of the stain are probably water soluble, a bit of vinegar or lemon juice might help with that. Flood and remove repeatedly, I would use a wet vac for that rather than a blotter or rag.
I suspect the dark stain is a reaction with tannin in the wood, much like ebonizing. That's tougher to deal with, but some oxalic acid or hydrogen peroxide might help. I think I would try the peroxide first.
That piece is almost certainly veneered. Avoid any deep sanding. I am also a bit worried about lifting the veneer, but then that would be repairable if a further can of worms to deal with.
The other thing is that if any method of lifting or lightening that stain works, it will still be visible. At best you make it light enough to inpaint with some stain and then overcoat the whole thing, equalizing the un-damaged portions with the repair. Bottom line is that its going to look different than it does now, and it will probably not look as the customer wishes. That style of teak furniture is typically very lightly finished, and in my view not appropriate for a dining table.
If the customer is willing to pay for a true fix that looks just like the old one, take the top off and re-veneer it with teak veneer. You'd also have to make new end pieces from solid teak I bet (those are why I think it's veneered). Not a hard job but expensive for such a little stain from a day to day sort of dining accident...