Theoretically, sure. Differences in the PHP and MySQL versions could certainly lead to different behavior. How likely it is that it only shows in settings not being saved, that is another question.
Do you get any messages in the browser’s console? Any Ajax requests ending in errors in the Network tab?
Is there anything in the error log on the server? Have you tested with WP_DEBUG?
You could also hook into wpdb’s query to debug which statements are actually sent to the database.