Yes, that's why feminists stressed that gender referred to the social expectations and limitations placed on each sex by the prevalent society.
It's easy to show that the vast majority of these are, indeed, social constructs: they vary wildly by location and era, therefore they aren't innate.
An inner sense of one's sex is easy to understand. Most people, having never questioned their society's gender constructs, find it hard to separate a "sense of one's sex" from "social stereotypes".
Then there's an ill-defined grey area in the middle, where some behavioural patterns are sex-linked. It's easiest to identify these in the area of mating-related behaviours and that, again, is rather grey because some people don't have any mate-attracting behaviours and some display those more usually associate with the opposite sex.
Going further, there are possible sex-linked behaviour patterns in other areas of life. Controlled studies to try and establish such patterns would be unethical, so we have no real idea whether they're all socially imposed or not.
From my extremely limited and selective reading of Butler's opus, she was originally talking about this grey area. A woman crossing and uncrossing her legs, for instance, is "performing [feminine] gender" as part of a flirtatious performance. A gay man walking with an exaggerated wiggle is "performing [feminine] gender" to communicate sexual availability. A man with his feet planted apart and shoulders squared is "performing [masculine] gender" to convey power, and so is a woman doing the same.
In feminism, gender performance is used to describe any gendered behaviour. I'm "performing gender" when I show up to an event with my hair and makeup done - I'm not sexually interested but, instead, am showing respect for the gathering's general expectations of female appearance.
Whether Butler took it further, I don't know, but it's easy to see how a superficial grasp of both concepts could lead to a hypothesis that all gendered behaviours are innate. Conveniently, this is exactly what people who never think about it think, so it's an easy sell.
From there it is a very small step to suppose that someone with an affinity for the usual behaviours of the opposite sex must really be the opposite sex, in character at least. Again conveniently, this dovetails with the popular intuition that the wiggling man "is a woman" and the dominating woman "is a man".
It's really bloody annoying!