Currently, "pay-stock/property" is a gift type option in Database View and not in WebView, and "stock/property (sold)" is an option in WebView and not in Database View. I reached out to BB Support and unfortunately, it isn't supported. We had entered a gift of "pay-stock/property" in database view and had applied it to a pledge payment and it somehow switched to "pledge payment" in WebView and not a "stock/property".
It would be great to have "pay-stock/property" as an option in WebView or "Stock/Property (sold)" as an option in Database View, so that the two gift types can be mapped or "talk" in both views.