Trans people say they identify as the opposite sex.

Humans have the phychological capacity to identify as anything.

Cisgender people are people whose gender identity matches their sex assigned at birth.

Have the people you refer to as cisgender verbally confirmed to you that they see their natal sex as a gender identity?

And they say everyone's gender identity is valid

It's not uncommon to see trans people disrespecting and putting false gender identity labels upon people who don't have gender identities.

Can GC explain what is wrong with what I said above?

I don't know you particular beliefs but many genderists talk as if we, the opposing side, agree with a genderist redefintion of "man" and "woman". If a "man" is gender identity to you, then you refer to a different thing than we do with "man", the same with woman, your "woman" and our "woman" are two differnet things. With woman I mean some like Buck Angel (an adult human female), with man I mean someone like Blair White (an adult human male), it has nothing to do with gender or identity.

Why do you say sex is not assigned at birth?

Nobody assigned Caitlyn Jenner with a penis at birth, male Caitlyn Jenner already had one. Nobody assigned Caitlyn Jenner with the sperm to father 6 children, Caitlyn Jenner's male body developed that naturally.

Why do you say people can't be born in the wrong body?

What do you mean with "wrong"? Can a body be "right"? I don't think a body be either as that would assume a brainless sperm with a y or x chromome is a moral agent who can do something morally wrong/right by merging with a egg.

Can you explain why gender identity does not exist?

It clearly does exist in some people just like cat identies exist in some people.

Don't cisgender people identify as/feel like they are the sex they were assigned at birth?

Trans people is probably the group who feel like their natal sex the most as they tend to be so acutely unhappy/dysphoric about it, had trans people not felt like their natal sex they wouldn't remember to take cross-sex-hormones. Regarding adopting natal sex as a gender identity, are you asking if cisgender people are cisgender or if non-trans people are cisgender? Only the subgroup who have adopted their natal sex as a gender identity are cisgender, the rest don't have gender identities at all.

They mean a dog or a fish also have a gender identity

I disagree, I don't think a dog or a fish has a concept of what gender identity is. They might know their natal sex the way trans people know their natal sex but as trans people evidently don't see their own knowledge of natal sex as gender identity, knowledge can't count as gender identity in animals either.

Is it okay to invalidate someone's gender identity?

It's wrong to lie about people's gender identies, speaking biological truths about sex is not invalidating gender identities though. Calling Caitlyn Jenner a man/he for being male can't be invalidate Caitlyn Jenner's gender identity as man/he refer to sex, not gender identity, to invalidate Caitlyn Jenner's gender identity we would have to lie about which gender identity Caitlyn Jenner has.