Good question, but there is no logical fallacy involved. The sentence is capable of being true or false, not in virtue of logic, but in virtue of facts. And, with the best of intention and without neat-picking, I tend to think it is very likely to be true - if you have not wondered why a certain beautiful woman goes out with a certain ugly man who is not only unattractive but also repugnant, you are not old enough.

Setting aside edge cases for the moment, let's consider the following statements:

No man understands girls' problems.
No man has sexual desire for men.
No dead person is bothered by pain.

The reason I think this question interesting is that it eventually asks what the word "understand" means: what does a person feel when he understands a situation?

When an object-word is understood, the listener's mind forms a mental image similar to the one which that word expresses. If that word is "fragrant" but the listener has no sense of smell, the listener will not understand it - i.e. he will not associate a noise with an olfactory sensation.

To understand how a steam engine works, a series of percepts are necessary, and mental images are the ultimate results.

It follows that mental images - feelings, sensations, desires, etc. - are most likely the ultimate results of understanding.

To understand another person's problem presupposes common feelings, and common feelings are the ultimate results. If a person has no sense of smell, he will never be offended by BO and will not understand why BO is a huge problem in the office. It follows that, if a person has no sense of smell, his opinion on BO is much less weighty than that of a person who can smell.

Similarly, the mother will never understand why the son is so attracted to a certain woman while the father virtually agrees with the son on every aspect. It follows that a mother's opinion on the son's girlfriend should not be taken seriously.

It follows that men's opinions on issues particular to women are very likely to be less credible than women's own opinions.

Being unable to understand does not necessarily imply lacking concerns for the sufferings of others. Civilized western men's attitudes towards women are nothing but admirable.

Being able to understand does not automatically imply compassion. An evil man is capable of exploiting your vulnerabilities for his sadistic pleasure; he may justify it by some karma yoga philosophy, which only reminds you of the kind of fine phrases used by a 20th-century saint to excuse himself for sleeping next to naked young girls.

"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.