Merely defining womanhood as something you identity as is obviously circular and doesn't make sense.
You are X if you identify as X
You are X if you identify as (if you identify as X)
You are X if you identify as (if you identify as (if you identify as X))
As we see, this doesn't work.
Sex as a bimodal distribution:
The most accurate model that includes all of humanity is to look at overall sex-characteristics, then for most cases it will be fairly obvious who belong in the male category and who belongs in the female category
The issue here is going from an accurate bimodal system to a binary categorization. You will always end up with cases where which of these categories they belong to.
The solution is to give up the idea of having some universal binary classification and instead have different rules depending on the situation. For example the rules for sports does not have to be the same as the rules for legal sex.
vintologi.com
Social sex vs biological sex:
For social interactions what matters is your presentation and this can obviously be changed. At least social transition is very much possible with regard to typical social interactions.
Whether or not transition changes your sex to actually go from biological male to biological female will depend on the exact definition, when you start and which medical treatments you get. This however is not very relevant for social interactions.
Things like womb transplants for trans females would be great in any case.
pinknews.co.uk
Ovaries for trans females allowing for pregnancy and eliminating the need to take HRT would also be a good thing.