There's no such thing as "gay ideology".
Being gay means you are sexualltly attracted to people of the same sex as yourself. It isn't a theory. It is self evident.
Gender identity is nothing more than a theory, or an idea. It is the belief that everyone has a gender identity, and that most people have a gender identity which matches their biological sex, whereas some people have a gender identity which conflicts with their biological sex, and that it is your gender identity and not your biological sex which determines whether you are a man, a woman or something else. It is the belief that being a man or a woman is something metaphysical, rather than biological. It is absolutely an ideology.
And it is an ideology which always, always, in its application, benefits those born male at the expense of those born female. When it comes to single sex spaces, sports, language, all of it confers an advantage on those born male and a disadvantage on those born female, regardless of what anyone calls themselves or how they identify or what pronouns they prefer. That's why an increasing number of people reject it.
So no, it is not similar to being gay in any way.
Being gay is a fact.
Gender identity is a belief system.
They are not the same.