The fact that the two contacts use the identical tooling tells me that all the key dimensions must be identical. But it turns out that if you look at the JST online shop, on the SHD housings, it says “use SH contacts”.
The contacts listed in the SHD datasheet are gold-plated (hence the G in their part number), while the ones on the SH data sheet are tin-plated (hence the T in their part number).
You do not want to mix gold and tin platings: the plating of the header and contact should be the same.
So if you get tin-plated SHD headers, then the wires you want to use are fine.