While there is no direct measurement of a "power level" in the Wizarding World, there is definitely evidence of it existing; although, it is not fully described where it comes from.
Magic in general in the Wizarding World seems to have an essence that can be concentrated and detected which at times seems almost sentient, at least in the materials that are used to make wands.
Mr. Ollivander describes this in his notes:
The best wands, he believed, had cores of immensely powerful magical substances, which were expertly enclosed in specially selected and complementary wandwoods, the result to be matched to an owner with whom the wand itself felt the most affinity.
Wand Cores
After much experimentation and research, I concluded that only three substances produce wands of the quality to which I am happy to give the illustrious name of Ollivander: unicorn hair, dragon heartstring and phoenix feather. Each of these costly and rare materials has its own distinct properties.
- Unicorn hair generally produces the most consistent magic... ... Minor disadvantages of unicorn hair are that they do not make the most powerful wands
- As a rule, dragon heartstrings produce wands with the most power, and which are capable of the most flamboyant spells.
- Phoenix feathers are capable of the greatest range of magic, though they may take longer than either unicorn or dragon cores to reveal this.
Quotes above taken from Wand Cores by J.K. Rowling Originally published on Pottermore on Aug 10th 2015
Wand Woods
Mr Ollivander believes that wand wood has almost human powers of perception and preferences.
Only a minority of trees can produce wand-quality wood (just as a minority of humans can produce magic). It takes years of experience to tell which ones have the gift, although the job is made easier if Bowtruckles are found nesting in the leaves, as they never inhabit mundane trees.
- [Acacia] When well-matched, an acacia wand matches any for power, though it is often underrated due to the peculiarity of its temperament.
- [Apple] They are powerful and best suited to an owner of high aims and ideals, as this wood mixes poorly with Dark magic.
- [Blackthorn] although it is undeniable that those who do so will enjoy the blackthorn wand’s prodigious power
- [Black Walnut] that of black walnut seeks a master of good instincts and powerful insight
- [Cherry] cherry wood often makes a wand that possesses truly lethal power
- [Elder] It contains powerful magic, but scorns to remain with any owner who is not the superior of his or her company
- [Larch] larch has long been valued as an attractive and powerful wand wood.
- [Laurel] I have known laurel wands perform powerful and sometimes lethal magic
- [Pear] This golden-toned wood produces wands of splendid magical powers, which give of their best in the hands of the warm-hearted, the generous and the wise.
- [Poplar] Here is a wand to rely upon, of consistency, strength and uniform power, always happiest when working with a witch or wizard of clear moral vision.
Quotes above taken from Wand Woods by J.K. Rowling Originally published on Pottermore on Aug 10th 2015
While wand cores and wand woods are "inanimate" objects they are originally produced from living magical beings. I do not think it is a great stretch to say that if certain creatures and trees can have a higher magical concentration that makes them more "powerful" that humans could have similar qualities. Although, I do not have direct evidence with supporting quotes to prove this.
This is part of the reason why blood status is a thing in the Wizarding World, blood purists believe that the magic in their blood is more concentrated or pure. We obviously see that this not the case as many half-bloods and Muggle-borns perform incredible magic. However, it may be that magic is able to concentrate in a particular individual.
At they very least it seems powerful magical artifacts (wands in the primary example) are able to amplify a wizard's skills and powers.
Further evidence to support a wizard's power can come in form of the spells they cast. Ones such as the Patronus Charm, Cruciatus Curse, Fiendfyre, are described as difficult to cast and require some for extra effort (typically some emotion). In general it seems the more effort a wizard puts into a spell the more effective it becomes. Some of this comes down to practice and skill, but other have seem to have an innate ability.
At some point this becomes self-referential. We know wizards have different power levels because the books describe it so...
