Learn English – Difference between family name, name at birth and first names

I'm not 100% certain what you're asking, but I can think of a few possibilities.

• As Kosmonaut commented, on most forms, an initials field is asking for a middle initial. If the form designer was sane, the field will appear between the first and last name fields, but you never know.
• The field could be asking for degrees and other initialisms that people put after their names – "John Henry Doe, OD, FAAO", "Lucy Smith, CBE", etc. If the field appears after the first & last names, and especially if there's a separate field already for a middle name or initial, then this would be the most logical explanation.
• As a remote possibility, the field could be asking for a title - Mr., Mrs., Ms., etc. Like the middle initial field, though, it would be bad design to have this after the name fields.
• If it's a long form, there could be places where you need to "initial" (give a quick sign-off for) some important clauses. I haven't encountered this with computerized forms, but it's standard practice with long contracts such as for buying a house.

More generally, using an article before a proper noun that doesn't have one built into it (as the United States and the Rolling Stones do) is one example of using a proper noun as a countable noun.

There are several reasons why we might do that normally. One is to say something like "there are three Johns in the group", meaning "there are three people called John in the group".

Another is to add distance to the identification; "I have a John Smith on the line" is a common expression for "I have someone on the line, who tells me he is John Smith, and that is all that is known about him". A similar is to report, e.g. "One John Smith is accused of the crime", emphasising that we have no further identifying details at present, and hence we are not stating precisely which person of that name is the subject of the sentence.

Another is to use a proper noun as an example of particular traits that could also be held by others (a type of synecdoche). "The next Bob Dylan" (a singer-songwriter from the folk scene who will repeat Dylan's success), "He's an Einstein" (he's very smart), "All Mozarts have their Salieris" (not really true even for Mozart and Salieri, but let's say we believed the film Amadeus was accurate).

Another, almost inverse to this, is to speak of the person or thing signified by the proper noun at a particular time, or from a particular perspective: "The London of a hundred years ago was a notoriously unhealthy place", "The John you know is not the John I know" (that could also mean you are talking of a literally different person, depending on context).

The above are reasonably standard, though figurative.

Another common variation is to jokingly make use of these forms, when one normally would not. If talking of a friend, we would generally use their name as a proper noun, because that's how names work in English, but since every person called George is "a George", and so on the form is logically correct, though not strictly good English. To use it of a friend could suggest that you have gotten as far as knowing it's a George, but not which one, or that George's are all alike and you've hence found someone who will have all the George-like qualities that George has. Both obviously are not sensible, but therein is the joke. Another variant would be if you were looking for George, and then spotted him. Again "ah, there's a George" would suggest that you'd were just looking for Georges generally, which again is not sensible, hence the joke.

All of these last cases are examples of deliberately bad English, used as a joke, rather than something that would normally be considered correct.

[A completely different case is when there's a word that is the same as a proper noun, but isn't a proper noun, of which some slang cases started as a proper noun and are hence sometimes capitalised.]