Yup, most gifts do not have preset origin game. And it's not something a "normal person" can identify too, unless they transfer the mon away (cause only then origin game would be identifiable, as region name).
Most gifts do not have preset origin game, and they normally just follow the received game. The ones that *might* have a preset origin game, are usually copies of mons belonging to real life trainers, like champions of VGC and stuff. That's why the Met location in Gen 6 is Route 7. It imitates the fact that it was a bred Tyranitar.
(In the wonder card, you'll see there's an Egg Level and Met level. If you look at my app near the bottom left corner, it's the 1 and 55 respectively)
edit: anyway, some of the info (in a more visual format) can be seen here: