I did some searching, an no, they are not built at the same plant.
While the 10-speed auto was a joint development between Ford and GM, each company assembles and produces the units within their respective manufacturing chains, so as to support specific models (Mustang, Expedition, 4x4, etc.).
Ford:
- Livonia, MI
- Sharonville, OH
GM:
- Romulus, MI
- Silao, MX
Different plants produce different units and case assemblies. The internals of the transmissions themselves, some of those bits might get produced by one third-party contractor. I didn't delve too deep into the topic.