Some of my classmates (and even some of my professors) confuse the terms universality and normalizing. Both of these are terms that have to do with the understanding of one's problem or issue as normal. They differ only in context, but it is this difference that makes them unique.
Universality is an idea that promotes an understanding of one's problematic behavior through the idea that: others have suffered in the same way, or similarly, in the past; are suffering in such a way in the present; and will most likely continue to suffer like they have in the future. This promotes the feeling and idea that they are not alone and that others have either been "cured" or have learned to solve their problem/cope with it in a way that they can live a happy life. This is one of Yalom's group theory therapeutic factors and makes a lot of sense when the immediate group can support the therapist or show their own similar behaviors/strategies on their own.
Normalizing is more of an intrapersonal topic. Instead of highlighting extrinsic factors, it describes the idea that the behavior itself is normal. All behavior is normal. Any mental illness is a product of all experiences of the past. This means that the psychotic is normal because he or she is just performing the actions that make sense to them given what has happened in the past/what happens to them every day. Carl Whitaker is very well known for accepting clients' behaviors as normal.
While similar in intent, these concepts are dissimilar in context. It is important not to mix them up because they are attributed to two different thinkers.