It is less about discrimination but about business needs, and also attitude/maturity.
First, there is business need. I have a friend who works in gastronomy, she is tattooed all over. Which, personally, I find repulsive, but hey... it's her body, not mine.
At work, she wears a buttoned-up white blouse covering her arms down to the wrists. She has to do that because she works in a profession with direct customer contact, and there are customers such as e.g. me who find tattoos repulsive.
There's a distinction between what you are allowed to do in terms of your personal self-fulfilment, and what you are allowed to rub into people's faces, especially at work (also, in some free countries self-fulfilment is limited insofar as you are not allowed to have any opinion, or speak out any opinion, or get any kind of tattoo anyway).
In a profession where you are engaged in direct client contract, it is your job to respect your client, and to keep the client comfortable. It doesn't stop at being tattooed (or covering them). You may for example, depending on what job you plan to work in, find that it is equally unacceptable to come in ragged, dirty clothes every day. There are things that you just have to comply with, or you won't get hired.
There are jobs where nobody cares, true. However, with the exception of a "typical startup", those are usually not the jobs that are very desirable.
In most "somewhat better" jobs, you must maintain a certain semblance of professionalism, and some particular overall appearance. Got a tattoo in a location that isn't visible? Or something small and inconspicuous? Probably not a problem.
But surely you wouldn't find it very reassuring if you went to a hospital and your doctor had a big skull and crossbones tattooed on his right arm, and F.E.A.R on the knuckles of his left, and K.I.L.L. on the knuckles of his right hand. That just doesn't work.
Would you trust a bank with your money if the cashier looked like KISS on stage? Or like Danny Trejo? What if the cashier had ragged clothes, unwashed long hair, hippie beads, and walked barefooted? Would you feel comfortable having your money at that bank? Do you think the majority of customers would? I don't think so.
Next, it comes to attitude:
he'd refuse a job in a company where he's asked to keep his future tattoos covered
To me, this shows a serious attitude problem, and it is in my opinion a clear sign that your friend is not ready yet to make such a serious, quasi-permanent decision as getting tattooed. Even moreso as he doesn't even know yet what job he wants to take later.
If you outright refuse to cover your tattoos when applying for a job where this is expected (and "normal"), you are just as ignorant as the people whom you accuse of being discriminating. Only just, you have a misled perception about who is the one asking for something. I mean, as someone who is eventually going to apply for his first-ever job, one shouldn't too much lose perspective.
Once you are in a position where yet another headhunter offers you a better six-digit salary job with even better benefits every 3-6 months, you can afford being picky, eccentric, demanding, and even obnoxious. As a beginner, that's quite a bit of hubris.
Let's be honest, you have to present excellent grades and have a distinguishing feat and give a good impression and be lucky to get a decent job at all, having no work experience.
Because, you know, there's literally millions of fresh, inexperienced applicants coming out of school every year, and ten thousands of these have excellent grades, and they all want a good job.
As closing words, it is my belief that most, if not all things in life work both ways. That's in particular true for acceptance and respect (and for rights and duties alike). Unluckily, in our modern society, people tend to think differently.
They want rights, but not obligations. They want respect, but don't respect others. They want acceptance for whatever they do, but don't accept that other people want to do something different.