The human mind has a natural tendency to make metaphors to describe abstract concepts. In English, one such metaphor is
"up" = good, in working order
"down" = bad, broken
Examples:
Is the mail server up?
The server went down yesterday when the hard disk crashed.
The dictatorship rose to power in 1969.
The regime fell as a result of the coup d'état in 1974.
In the heyday of the motor industry, things were looking up.
However, since the collapse of the motor industry, Detroit has gone downhill.
He hit rock bottom when an alcoholic binge landed him in the emergency room.
He has since rebounded.
Management approved proposal A, but gave a thumbs down to proposal B.
Combine that with the arbitrary convention to orient maps with north pointing up and south pointing down, and you get an idiomatic expression "going south", meaning "turning bad".
Note that these metaphors are not always generalizable. For example, going uphill is not the opposite of going downhill. ("Uphill" usually refers to some kind of struggle.) Similarly, "going north" is not a commonly accepted expression. However, it is possible to say
Profits are expected to be north of $3 million this year.
… meaning "above 3 million dollars".
"Form" and "function" are often contrasted in English.
Contrasting or relating these goes back to Aristotle. Most concretely, "form" just means shape. More abstractly, Aristotle explained how it's possible for a thing to change and still remain itself by saying that its matter stays the same as it takes on a new form. To this day, when we want to talk about ways in which the "same" thing—like a word—can be different, we speak of the "forms" of that thing. There are many different "forms" of democracy, "forms" of web site, "forms" of music, etc.
Aristotle also talked a lot about the "function" of the various parts of plants and animals. One of the central ideas of his philosophy is that form and function go together: the forms of the parts of organisms are explained by the functions they serve. In other words, the role or purpose—the function—that a part serves in the life of the organism explains its shape and design—its form. Some teeth are sharp, for cutting; some are shaped for grinding; the fingers are jointed and placed so they can work together to manipulate objects; etc.
"Form follows function" became a well-known motto of 20th-century architecture. The idea is that the design and shape of a building—its form—should reflect its function, and not useless ornaments or traditional frills.
So, to answer your question, when people speak of "form" in grammar, they can mean a lot of different things. The "forms" of a noun are the specific words that indicate case and number: for example, in English, "I" and "me" are sometimes called "forms" of the same word, as are "pen" and "pens"—the singular form and the plural form. English verbs have up to five forms, illustrated by show, shows, showed, shown, and showing.
All the forms of the same word are sometimes called its morphology—from morphe, the Greek word used by Aristotle that was translated in ancient times into the Latin forma. Syntax is the way the sequence and forms of words are chosen in a sentence, sequence being easily considered another kind of "form". Some people speak of sentence forms: declarative, interrogative, or imperative; or simple, complex, compound, and compound-complex. The -tion morpheme takes one "form" in position and another "form" in transmission. People call upon the word "form" in grammar to mean pretty much any way that a single, underlying element of words or sentences can appear differently in different situations.
In general, grammar is concerned with how the forms of words and sentences express meaning, distinct from the meanings of the individual words—for any reasonable use of the word "form" in regard to words and sentences. For example, the difference in meaning between "I see him" and "He sees me" is purely grammatical. Both sentences have "the same" words but the words are in different forms and the sentences have different forms, resulting in different meanings.
Best Answer
In English there exists an interjection -- "For shame!" -- which expresses disapproval.
The parenthesized comment explains that the previous "for shame" is not that phrase but simply a way of wording "because of shame". But then, the author could have just written "because of shame" and omitted the explanation. So it is clear that the author wrote "for shame" followed by an explanation deliberately, in order to play on "For shame!", in order to sneak in an expression of his or her own disapproval of the situation.
"Which is a shame" is simply an expression that the preceding situation is unnecessary, wasteful or regrettable in some way.
Note that in "for shame", as used by the author, the "for" means "because of" (we have to take the author's word for it). But in the "For shame!" interjection it doesn't have a meaning. The expression is an idiom without any literal interpretation according to any general rule of grammar. We can hypothesize that "for" plays the role of a special particle for starting such interjections, and it may simply have been copied from expressions such as "For God's sake!" in which it does play a normal grammatical role (literally saying "for the sake of God"). Another expression with such a "for" is "For crying out loud!".