The closest we can get to "shapeshifting" with today's technology involves a multi-pronged approach consisting of extensive cosmetic surgery, disguises, and acting ability.
We're nowhere near implanting octopus skin, building nanomachines, genetic sculpting, or even implanting anything more complex than a pacemaker long-term. We can clone people (even if it's never been done publicly) and we can make "designer babies" although our current customization level is rather primitive (eye color, hair color, skin tone, basic stuff). We can't alter someone on a biological level in a reasonable manner once they're already born.
That said though, surgery can and has done things like:
- Complete face replacements
- Partial skin replacement
- Hair implantation (commonly used against balding)
- Height adjustment (change length of legs)
- Change someone's gender appearance
- Head/brain transplant might be possible but it's rather iffy
This means, that provided someone has boatloads money to burn, can find unscrupulous doctors, and is willing to undertake significant personal risk, they'd be able to transform themselves into an unrecognizably different person.
After surgery, the rest of the shape-shifting can be done with mundane disguise work. Clever makeup application, well-made glued wigs, and a new wardrobe to sell the look can make a world of difference.
Finally, to sell a disguise or new appearance, the shapeshifter would need lessons in acting and potentially extensive physical therapy if the surgery has changed their kinesthetic layout (eg if they shortened their legs).