We only know that Flamel owns a powerful magical object.
Hermione describes Nicolas Flamel in chapter 13 of The Philosopher's Stone:
Harry and Ron barely had time to exchange mystified looks before [Hermione] was dashing back, an enormous old book in her arms.
'I never thought to look in here!' she whispered excitedly. 'I got this out of the library weeks ago for a bit of light reading.'
'Nicolas Flamel,' she whispered dramatically, 'is the only known maker of the Philosopher’s Stone!'
This didn't have quite the effect she'd expected.
'The what?' said Harry and Ron.
'Oh, honestly, don't you two read? Look – read that, there.'
She pushed the book towards them, and Harry and Ron read:
The ancient study of alchemy is concerned with making the Philosopher’s Stone, a legendary substance with astonishing powers. The Stone will transform any metal into pure gold. It also produces the Elixir of Life, which will make the drinker immortal.
There have been many reports of the Philosopher’s Stone over the centuries, but the only Stone currently in existence belongs to Mr Nicolas Flamel, the noted alchemist and opera-lover. Mr Flamel, who celebrated his six hundred and sixty-fifth birthday last year, enjoys a quiet life in Devon with his wife, Perenelle (six hundred and fifty-eight).
Flamel is not known to have created his Philosopher's Stone.
Hermione asserts that Flamel is the only known maker of the Philosopher's Stone, but the passage she cites to support her claim lists him as owner, not maker. As we know from The Deathly Hallows, the ownership of powerful magical objects transfers from person to person, down the centuries, through means ranging from gentle to violent.
Flamel is an alchemist, but he is not necessarily a wizard.
The Harry Potter novels never explicitly refer to Flamel as a wizard. Albus Dumbledore's chocolate frog card mentions "[Dumbledore's] work on alchemy with his partner, Nicolas Flamel" (Philosopher's Stone, chapter 6), and Hermione's library book states that "the only Stone currently in existence belongs to Mr Nicolas Flamel, the noted alchemist". Flamel does not seem to have any other claim to fame in the wizarding world, apart from his work in alchemy.
Alchemy has a long history with the (necessarily non-magical) people of the real world, so it's reasonable to suppose that non-magical people in the Potter world were similarly interested. In Deathly Hallows (chapter 16), a quote from Bathilda Bagshot's A History of Magic dates the signing of the International Statute of Secrecy to the year 1689, when Nicolas and Perenelle had already been using Nicolas' Philosopher's Stone for centuries. At the time of the Stone's creation, there was no law preventing wizards and witches from sharing magical knowledge with muggles. Nicolas could have been a brilliant researcher, without an ounce of magical ability of his own, who used the magical resources available to him at the time.
An unequal partnership?
Although Dumbledore — a brilliant wizard — worked with Flamel, the partnership does not imply that Flamel was any great talent, nor even a wizard. Dumbledore had a lifelong fascination with conquering death, and a willingness to go to great lengths for the greater good. The friendship could have begun selfishly with Dumbledore's determination to study a Flamel family heirloom.
Given the wizarding world's prejudice — during Harry Potter's school years — against anyone not a magically-talented pureblood, publishers would face pressure not to speak well of anyone who failed to meet these ideals.