Are parentheses denoting where a character is from necessary when a character has an unusual name that probably only belongs to them?

If a character has a unique enough name, they don't need to have a parenthesis suffix, strictly speaking. Ideally, the parentheses should only be applied to character names that are used by more than one character so we can tell them apart. Still, we're hardly paragons of tagging perfection, so there are a number of exceptions here and there on the site. Sometimes, people like to add on the suffix anyway because they feel it looks better and more consistent.

The presence of suffixes is necessary when the name is not unique. That does not mean that the usage of suffixes should be avoided when the name is unique.

How does this work in relation to it's own franchise?

As I see it the trend seems to be, character tags for iconic characters are the "default" and the franchise tag is in parentheses
sonic the hedgehog vs. sonic the hedgehog (series), or hamtaro vs. hamtaro (series)

Though it's the opposite for artist OCs. Also understandable given their lesser known status

Going off that, I tried to have hello kitty changed to the main character tag as opposed to hello kitty (character), but something about the current aliasing won't let me.

It Depends™