Installing watch faces is not as trivial as it sounds. When clicking the install button on the Play Store website, you will have to select the correct device first from a dropdown first.
On the phone, the Play Store app can be even more frustrating. Depending on if a watch face has a phone companion app or not, there might not be results visible until you switch from “This device” to “Watch”.
To make sure that you install the watch face on the watch, click on the little triangle on the right side of the Install button and tick the checkbox in front of your watch device before clicking Install.
If your watch device is not available from the dropdown, open the Play Store in a browser instead of using the app.