There are lots of ways an immortal can hide, even in this day and age.
The most obvious is, don't get famous. If fame happens upon you (save a kid from a burning building or something), let the furor die down, then do a perfectly natural move to another state or country. Forge a new identity there.
That aside, I imagine they'd do it the way criminals do; have a bunch of assumed identities ready and waiting. That would be a good idea to always have on hand, as well as a go bag with money and necessities in case the immortal needs to split rather quickly.
Of course, to get such IDs means the immortal will make contacts in the underworld, so it's not beyond the pale that they can also find a hacker to help them erase their footprint in their past identity and become someone new. Digital trails are only as good as the databases that hold them.
Hair can be dyed grey; they sell silver hair dye now. If male, hair can also be shaved bald, in which case no one will notice hair color. Women and men both can also wear wigs. Men can grow a beard to age themselves and change their look, while women can use makeup tricks and, as the "grow older" just stop using it (women who don't use makeup are often accused of looking old and/or ill). Clothing also figures heavily into our perception of age--if the immortal always carefully dresses in the type of clothing we expect for their age group chances are that people will question their age less. Also, claiming some Asian genetics could help since there is the impression that folks from that area of the world never look their real age.
And never forget how someone acts. The teenage slouch, the middle-aged ramble, the swift pace of a young urban professional--you have an idea in your head already of what these things look like because you've seen them before. We all have. And immortal can "age" by aging their vocabulary and actions. While in their "twenties" they favor beer, "thirties" wine, "and in their "forties" they switch to whiskey. Maybe not. But small touches like that added together can make a person a chameleon.
Done well, and lack of wrinkles will probably just get them envy and some head shaking, so long as they don't push their identity too far past the 45-55 year old window, depending on the character of their face. Babyfaces may want to cut out earlier, stronger features can probably pull off later. Either way they can probably pull off a good 40 years in any given area, more if the "elder" persona becomes a cranky hermit and people now are interacting with their "nephew/son/niece/daughter" who have agreed to care for them as they age.
Being unseen isn't too hard either. Alaska is still a good place to vanish to, sometimes for decades. Anywhere rural is a great place to vanish. The locals might pay attention, but overall no one important pays attention to the locals because they assume they're unintelligent, bigoted, and given to believing conspiracy theories and fake news. Plus, if you make friends easily and go to the local church and act devoted, they'll pretty much leave you alone. Do them a solid and they'll actively close ranks around you, no matter how weird you might be.
And don't forget how much of this country (if this is set in America) is run down and deserted. There are ghosts towns, empty houses and businesses, old sewer and rail lines, and other things just all over the place. Heck, there was a runaway kid who lived at the bottom of an elevator shaft for a couple months, and the only reason they were ever caught was because they liked to fry hot dogs and people noticed the smell. Any immortal who learns how to survive on very little could wander into a place like that and not be seen for years.
Becoming part of a criminal/minority (not in the racial sense)/endangered group (motorcycle gang/impoverished farmer/homeless) can make a man vanish instantly. People actually work at not seeing those sorts of folks, either out of fear or a latent guilt they don't want to feel.
And then there's always another country. Developed nations might be plastered with Big Brother levels of tech, but much of the world is not. Even those places that are still have less policed rural areas. The entire world is full of jungles, deserts, ocean islands, sprawling forests, and more. The canny immortal could do an excellently credible job of falling off the planet if they really put their mind to it. Even one strapped for funds could sneak into Mexico and then just make a beeline for the nearest South American jungle.
Another thing to keep in mind is that the forensic levels of police are not nearly what you see on TV. DNA is not used as often as it should be, and most departments have no budget for shiny, crime solving tech, even when it does exist (and generally TV makes most of it up). So an immortal doesn't have much to worry about unless they actively have someone investigating them.
As for item 2--legal channels--is your immortal willing to spend a fortune and wait decades, then fight whatever decision is made for another decade or so, all while dodging people who want to own, harm, or end them in the meantime? If the answer is no, then s/he may want to skip legal channels altogether.
If you want legal channels your best bet is to set up some shady government agency who know their identity but keep it on the down low. In that case your immortal might live at a special government site where no one cares about their lack of aging, with the drawback that, in return the government gets...something, whatever it is they feel is useful, as well as semi-intrusively keeping tabs on the immortal's behavior and/or laying down rules they must follow.