Recommendation for a projector module or approach?
As written in the comments, the linked projector is a DLP  projector with 640x320 pixels. Of course, that is expensive. It is a complete pixel display, which can display anything (with the max resolution of 640x320). That is way more, than the projector watches.
The watches often use one of two possibilities:

A bright 7 segment LED display, which is projected to the wall via a lens. The seven segment display needs to be bright enough, depending on the intended brightness of the projection and how big the projected image should be.

An LCD display, which is illuminated from behind by a bright LED. That is then also projected to the wall with a lens. I would guess, that this is, what the linked watch is doing.


How to build that yourself? You can always salvage one of these watches. But when building it yourself, you should also follow one of the above possibilities. Which one depends on your requirements.
Using a standard 7 segment display will be the easiest. Though that might be not bright enough. You could look for especially bright 7 segment displays and test, if that is enough for you.
The second possibility would be more hacky. Most LCD displays, that you can buy, have a LED lit plane in the back as a backlight. You would need to remove that plane, so that the display gets transparent. Also you most likely would need to rotate the polarizing filter on the top of the LCD, because as standard a driven segment of the LCD gets more opaque, while you want it to get transparent. A new filter (up to a few cm) can be salvaged from the passive 3D glasses from a movie theater.
In both cases you install the display inside a tube together with a fitting lens. Look, what you can get from your location and then experiment with it.

can you link to parts

No, we are not really doing that here, because that is really dependent on your location, on your price- and other requirements. Also such part suggestions are usually very fast outdated. But you can look at the electronics shops, that you have at your location and search there for the general description of the parts (for example &quot;7 segment display&quot;).