Thanks for the suggestion, but that's not really the same kind of mascot, at all.
Everyone has one, they belong normal families, rather than a hero (or somehow special person) or hero (or special) team, the main character is a not so cute one...

Whoops, nice catch =P

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A just ruler amongst tyrants

Since the mascot creature generally accompanies one of the main characters, it shows up a lot, and they can almost always speak, too (sentient). For those reasons, I feel like the mascot is basically getting as much attention as a main character would, and should essentially be a main character, as in, it needs to have a developed personality and serve as much of a purpose as any other main character does (at least for the sentient mascots). When done like this, I quite like them.

But, I feel like using using such characters as nothing more than cute pets (as in, they don't do anything else but look cute) is cheap--if that's all it's going to do, don't make it sentient, and don't give it so much attention.

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Manga Cover Database

A mascot character doesn't have to be unique in existence. It can be a common pet with a quirky personality like the cats from Nyan Koi! or Sumomo from Chobits.

^ Punch from Adachi mangas. So adorable xD

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I usually tend to like the mascots. Most of the time they're pretty cute. Well, from the ones I know anyway.

I think their alright. I love..love..Kero from Cardcaptor Sakura, but then theirs Happy from Fairy Tail who I just don't care for that much.

Well I can see what you mean, with Sumomo. She's not, technically, a mascot, but... Yeah. She's, more or less, a mascot in all but name.
Nyan Koi, however... No. There are no mascots in Nyan Koi, nor any being who is similar.

I usually find them REALLY annoying, especially if they're supposed to be cute. Don't get me wrong, I love real-life cute animals. But these guys grate on my nerves.

I vote for #4, since they're tolerable if they actually do something. But if they have to be there for a functional role, I'd prefer they not be cutesy.

Depends on what you consider to be mascots. Nyamsus and Tama recur enough that I consider them as such.

Ugh, it just pisses me off and I feel like slapping the author on his/her face when an animal is added with the thinly veiled purpose of cute.

...
Recur enough? By that logic you might as well say that Misty and Brock, in Pokemon, are mascots.

OO ... flat-chests are loved by the Devil himself.

I'm generally okay with mascot characters that don't affect plot, because then they become more than what they are supposed to be, IMO. It should remain the pet, and if in a comedy should provide much, if not most, of the hilarity that the MC is involved in. If not a comedy it should be a character that the reader feels attached to and interested in, so it could be cool, smart, mysterious, outspoken, etc.; some unique character trait to make it a part of the reading experience different from a regular extra or minor character. I couldn't care less for cuteness, and their mere presence doesn't elate me either. I don't think they are wholly unnecessary because it all depends on how the author deals with it. I don't prefer the mascot to "serve a functional purpose" because that's expecting too much of a mascot, again IMO. So, I'm okay with them, and will reserve judgment for when I actually see how it is presented in a story.

But they aren't animals-like creatures or something similar.

I never said that frequency of appearance is enough to make you a mascot. On the contrary, I was saying that being a mascot isn't about that.
There are certain things that make mascots, be mascots. How frequently the character appears, is not one of them. It'd be weird and bizarre to have an infrequent mascot, but they'd still be mascots. Frequency of appearance is not a defining characteristic ...and I don't see how Nyamsas or Tama possess any characteristic, which is typical of a mascot.

Also: Nyamsas and Tama aren't animal-like creatures. They are cats. They are about as far from being animal-like creatures, as you could possibly get.