There are various potential causes of anemia, but when iron deficiency is the cause, anemia is a late-stage development of the deficiency. The body begins to remove iron from storage and from other bodily functions that it deems less essential in order to keep producing red blood cells. Even if that means robbing Peter to pay Paul. Think of it as "battery-saver" mode on your phone: it can get things done, but not at full capacity, because it's trying desperately to economize on resources that it knows are critically limited. This is why iron deficiency, even without anemia, can cause so many problems with hair (falling out), skin (dry, slow to heal), metabolism (generally also slower), mental focus problems, anxiety (because the body, on a primitive level, recognizes the challenge to oxygenate as a legitimate sort of emergency), and so, so, so much more.

All of this is why non-anemic iron deficiency can be so symptomatic. Some researchers even suspect that, while the blood has adequate oxygen in pre-anemic iron deficiency, not all of the body's tissues necessarily do (hence, for example, the brain fog and anxiety that can also come along with iron deficiency).

Given how much iron is lost through menses, and especially how much blood has already been lost by the time of perimenopause (or is even being lost in excess during perimenopause), it would have made sense for her doctors to check her iron too. :(

Once menopause comes, however, iron loss ends, and (depending on their diet) a person's iron levels can gradually recover, even if they aren't supplementing.

(All of this is my non-professional understanding of the situation though, so ask a doctor if you think you're iron deficient and want a test. Actually, ask several, and preferably female ones. Don't supplement with iron on your own though; it can be fatal if you don't need it.)

EDIT: I should clarify that iron loss doesn't entirely stop at menopause; it's monthly iron loss that stops during menopause.