Why do people use all 3 components in their gender pronouns?

Not all people want others to use the standard pronouns, whether masculine feminine or plural. Other possible pronouns exist such as ze, hir, zirs etc. Many people are unfamiliar with the usage of those and what the object and possessive forms are, so it makes sense to specify all of the ones you want to be called by in each circumstance.

There are now a number of additional pronouns, mostly used by people who identify as non-binary (neither 100% male, nor 100% female, or a third gender). Most people are not familiar with them, so don't know how to use them when only one is presented. If one pronoun is "xe" or "chis", what are the others? Can you work out the rules without being told?

The generally accepted pronouns are not regular, in fact they're highly irregular so they have to be memorised. Most people have not had enough exposure to these additional pronouns to have memorised them and be familiar with them.

This is why two, three, or even the full set of five pronouns will be mentioned.

There are badges and T-shirts on sale, each displaying the 3 forms of a pronoun, e.g. "she/her/hers". So that sort of set of 3 forms is what people display (those people who do that sort of thing), so that's what people see, so with positive feedback that's what's become popular.

Why those badges/T-shirts are made like that, I don't know -- that's a different question. It could be argued that another form, the -self form, should be included (e.g. himself, herself, xyrself). Then again, it could be argued that not only is that -self form needless, so is the -s form (e.g. hers).