Freelancer and self-employed workers don't really have an employer, but clients. Since it could be in their interest to keep the client list secret, how should they proceed when applying for jobs as a potential employee? Putting the company name, address and person of contact would be opening a door for other competing freelancers. Not putting anything, would mean closing the door for the job seeker.
I have some clients I keep confidential, and others I have no problem naming publicly. My employment history is a mix of 'full time' and contract, and I tend to name the 'full time' employers, and keep the 1099 relationships a bit closer to the vest. This is done to protect them more than it is to protect me.
Much of my work since 2009 has been 'reverse engineering', and some of what I've been reverse engineering is software that is anywhere from 10 to 20 years old. Some software license agreements demand that the user not 'reverse engineer' their product. Since I am not usually literally decompiling the code, but only analyzing the resulting output files, and given that vendor support and marketing for the product ended eons ago, it's hard to tell whether the vendor would care. Discretion is simply a good idea under the circumstance.
Similarly, if I am working on a 'custom' system that supersedes a commercial product that a user has installed, the existing vendor might be interesting in knowing they will soon lose this customer. Therefore, I don't name names.
I was working for an employer on a full-time basis that had a policy that the employees would not discuss the company in their personal social media sites. Given the business they were in, I could easily see the reason for it. I quit working for them due to what emerged as excessive demands, however I've honored their policy ever since. I don't have any personal animosity from the experience - they wanted things I couldn't do, I left, and life goes on.
Other companies I've contracted with no longer exist, therefore naming them isn't going to do any competitor much good. I can't even recall when I've been in a situation where someone else wanted my client - the kind of jobs I pick up are usually things no one else will touch.
It would depend upon the requirements of the potential employer.
If the company in question has never hired, or only rarely hires, freelancers, then they may not understand the desire of the freelancer applying for a position to keep his/her clients confidential. They may request references who they can contact and this will mean listing your clients.
If the company does have experience in hiring freelancers and they are willing to "risk" hiring you without contacting references or only after testing your abilities prior to an offer of employment, then you might be in a better position to negotiate.
It's up to the discretion of the hiring company how they decide to hire potential employees. While you can discuss with them your concerns about revealing certain information,ultimately they will decide what they need from you, the applicant, to comply with hiring processes.
Have a brief discussion by telephone (as an email or text can be misconstrued) with someone in the target company's human resources and discuss your concern. They will then let you know what is required and what is not.
It is somewhat a game of salesmanship.
First, it should be explicitly understood with your clients that you will use your relationship with them for your own marketing purposes. If the clients object to this, it is much better to know this early on than after the fact. If they want your relationship with them to be confidential, then it is up to you whether to accept this or ask for additional compensation due to this.
Once the expectation is set, then you must also explicitly ask to use individuals at that client to be references, as well.
Once all of that is established, then you should be selective as to who you reveal to your prospects. My rule is that you should tell them of clients you have that are one level above the prospect.
As an example, I was a freelance video engineer in a major market years ago. I was chief engineer of one of the "Big 3" production houses for some time before striking out on my own. I got a lot of work from the second-tier companies because of my reputation there, and eventually a lot of work from one of the others in the "Big 3."
When it came to the "third tier" plants (usually one or two people working independently), telling them I worked for the "Second tier" companies worked well, because that was where they aspired to be, and they knew I could help them up that path. Telling them I worked with other "third tier"companies didn't work well, as they were viewed as competitive threats.
As for "competing freelancers" - if the client knows about you, then the client most likely knows about them. You have to beat the competition the old-fashioned way: Better performance at a better rate.