Hi @simbian,
At initial inspection, it seems to be a CSS conflict issue. However, I want you to take a few steps on your side and then update me with the results.
1. Clear all the cache and then see if the issue is gone or not.
2. Do a theme conflict test. Switch to a WordPress Default theme and then see if the date field shows up properly or not.
3. Finally, do a plugin conflict test if the above two do not work. Deactivate all the other plugins except User Registration and then check if the date field shows up correctly or not. If it does then activate one plugin at a time and then keep checking at the same time. You will come to know about the conflicting plugin.
Lastly, if all three does not help then let me know and I will get back to you with a different solution.
Regards!