The time of women's puberty and menopause is a key health metric for a given population. If it's moving up or down something's likely to be very wrong (e.g. exposure to pollution, hormones in the food/water supply). It would be negligent not to track it for health reasons.

There seems to have been something shady going on here. I won't comment on that, but the government tracking this isn't per-se strange or unwarranted.

Yes it is. Do you think the US doesn't maintain similar statistics based on age? The data including more fertility indices, such as health issues or similar would be another story.

I'd bet dollars to donuts that insurance companies in the U.S. are tracking similar data.

Of course it is. Governments collect vast databases of statistics about all manner of attributes of their population as a matter of course. They even keep track of things like which sexually transmitted diseases people have, and how much money they earned.

They’re not sure who owns the database.

Depends, no? If the translation is "women between 18 and 40", it's harder to be upset about it, because US census data clearly has similar information.