"A woman is anyone who identifies as a woman" is a definition, not an argument (it defines what "woman" means). So it cannot be fallacious. But it's circular*, which means it's not a very useful definition. Definitions are supposed to tell us what things mean, but this doesn't do that.
As an analogy, consider: "a flarposjprpahopg is anyone who identifies as a flarposjprpahopg". This provides no insight into what "flarposjprpahopg" means, other than it's something that it's someone can identify as.
*: One could argue, technically, it's not circular, because it's possible for someone to identify as a "woman" or a "flarposjprpahopg" regardless of what those words mean. But for it to make sense for someone to identify as such, one would need to know what the term means, which requires a definition. If the definition only entails identifying as such, it would not make sense to identify as such.
Definitions are not objective
If someone takes the above definition (or the definition of "A woman is an adult human female" that is typically used by anti-trans people, with "female" referring to sex), as an objective fact of reality, that would be fallacious, because definitions of words are subjective.*
The definitions we choose to use may have consequences for others, such as whether we are inclusive and considerate towards trans people, in the language we use and in the systems we've built around that language (e.g. gendered bathrooms and changing rooms and prisons), whether they're allowed to access scientifically-verified effective medical care, etc.
And yes, while the definitions we use are subjective, this wouldn't excuse someone choosing to not be a decent human being by using a definition that results in harm to others. On the contrary, it puts a lot more responsibility on each person, to choose a definition that doesn't result in harm, or that results in as little harm as possible.
*: Although one could also state a definition roughly (but not strictly speaking) as a fact, to say that's what the definition should be, to propagate that definition. This would not be fallacious.
Replacing "woman" with "female"
Yes, that could work, although arguably not as you did.
If we assume commonly-used (progressive and scientific) definitions of words, "female in gender" is somewhat nonsensical. "Female" (as a noun) refers to sex, and it doesn't make much sense to say "sex in gender". If you treat "female" as referring to gender (i.e. a woman), you'd be back to a circular definition.
We could, however, use a definition such as: "A woman is someone with an internal sense of being female". This would have woman (gender) reference female (sex), so it would therefore no longer be circular. It also seems to match the relation between gender and sex, as they're typically defined by pro-trans people and scientific organisations (e.g. the WHO).
Additionally, this definition would avoid the common confusion/ambiguity with what it means to "identify as" a woman. Pro-trans people typically mean this as having this internal sense of being female (and saying you're a woman is just communicating this internal sense to others), whereas anti-trans people typically take this to mean someone simply needs to say they are a woman on one occasion.
Note that having an "internal sense of being female" does not mean someone believes themselves to be biologically female nor to have genitalia that they don't have, but rather that they have a female brain, roughly speaking, along with the associated psychological traits and characteristics.