Your definition of "wealthier" is stupid, though, because it relies on a measure that doesn't actually apply to the person in question. A share's price is just "the last sale price of the same type of share", and that is multiplied by the person's shareholding to get their "wealth". However they don't have any more money by this happening - it's just a huge assumption that the share price would stay the same across all their shares if they sold right now.

The time we actually know what money they would make, making it concrete for tax, is when they sell. And we already do this.