The key point of your question is this:
"[...] all the registrations have the registration (ID or Code) and they
are very important for customers to use to relate a
register with the other [...] "
Yes, they are certainly important ... to the company that created and uses the system . First of all, it is a nuisance for a user to have to remember a code, however short it may be. Second, in terms of identification data, nothing is more important to the customer than their own name. Certainly the clients of these registrations you mention would prefer to use their names to relate. Do not believe? Ask the customers. Or do an empathy exercise and put yourself in their shoes with some service you use (telephony, entertainment, etc.).
If the code is something really necessary to the business, ease the life of the customer by putting it on a card. This is not a brilliant idea: filmmakers have been doing this for a long time (in the days when they still persisted). If you need a code, let the customer define it. This is called a login, and email is usually used not only by the contact, but by the uniqueness.
Well, now considering the user not the client, but the person who deals with the client and uses this interface for some relationship, the code is even less necessary. What this user really needs to know is the name of the customer, perhaps your email and other data of the user's identity or their situation in the business, depending on the problem's domain. In fact, in a well-done system, the lines in this table already demonstrate the action that the user needs to take in relation to that customer, and if they do not even make the phone call (an example of telemarketing) even while displaying the phone. Why would the ID be needed in addition to its use in the database tables, ie why would it be necessary in the interaction?
What usually happens is that such an ID is so important to the business that it pushes down the client and the user that serves this one, but almost never such a code is really necessary. See what UX doing is just asking these questions (difficult and sometimes bothersome to anyone who is already accustomed to such a "desktop apps world").