A friend of mine is an experienced researcher in early career. We are surprised to learn that she never get invited to referee for any journal. I once rejected a referee request and referred the editor to her because the manuscript matches better with her expertise, but still, she did not get invited. I recently learned that this happens more than once: some one else also recommended her for referee but the editor did not take the advice. It becomes a little bit concerning: is she somehow blacklisted or what? What is really happening here?
Your friend might be on a specific editor's blacklist, if she has engaged in poor reviewer behavior with that editor (submitted very short or low-quality reviews; accepted reviews for which she has a conflict of interest; leaked ideas under review before the authors have publicly posted a preprint of their paper; etc). But it's unlikely that she's on any kind of journal-wide blacklist (in some systems like Elsevier, editors have the ability to assign reviewers "scores" visible to other editors, but almost nobody bothers doing this) and extremely unlikely that she's on a field-wide blacklist---these things don't formally exist, and she wouldn't be shunned by her research community unless she is infamous for a history of gross misconduct.
I can't speak to sexism as a potential cause as I don't know the statistics---I wouldn't be surprised though if women are asked to do more than their fair share of reviewing (it's behind-the-scenes service work, with no recognition or prestige, after all).
The most likely explanation, I think, is that she's not yet visible enough in her field to receive a lot of reviews (the OP mentions she is early-career). Even if you suggested her as an alternate reviewer for some papers, perhaps the editor already had enough alternate reviewers in mind whose expertise more closely aligned with the topic of the paper, or with whom they already have an established relationship of trust.
I don't think your friend should worry too much, but she of course should think about ways of increasing her visibility, so that her name naturally comes to the mind of any editor handling papers in her area of expertise. In addition to making sure she is easily findable by editors (setting up a personal web page with list of recent publications, Google Scholar profile, etc) she could also volunteer to serve on papers committees for conferences in her area. Obviously, she should also just keep publishing: being prominently cited by a paper under review greatly increases the chance of being asked to review that paper.
It's quite possible the editor never read the recommendations. Some journals invite more reviews than they need. If some reviewers decline, but enough reviewers return reviews for the editor to make a decision, then the editor will not consider inviting more reviewers. If the decision is to reject, it may be made after only one review is returned.
Never getting invited is a bit odd, but could occur by chance. I only get 33% of my fair share of invitations, even though my reviews are faster and more detailed than the other reviewers. Most journals in my field have never asked me, other very similar journals ask repeatedly. I have received no invitations for many months, and then gotten two from different publishers on the same day. Don't overanalyze it; don't be concerned.
Most publishers have a form you can fill out indicating your research fields. If you complete this it might increase your chances of getting papers to review. In my experience, it is not necessary to publish with a journal or publisher in order to get asked to review.
I kinda depends very much; journals/publishers usually keep a list of referees and their fields of expertise, so the first step of the editor may often be to query the database of people who have already refereed for the journal. Thus, the system is designed to give the refereeing jobs to those who have refereed before.
If you are “on the list” of go-to referees, you get asked all the time. It can take a while to make it to this list, but if you’re there and do a decent job, the non-linearity of the system can easily overwhelm you.
Just a single aspect to respond to a part of the question. The only reason why as editor I'd ever "blacklist" anybody would be that there's something wrong with their reviews, either unfair and useless, or promised but never sent. In general I'm happy to have an as large choice of reviewers as possible, so surely nobody who has never made a bad impression reviewing (see above) would ever be "blacklisted".