Something living has a personality or persona of its own. Similarly, is there a word that can be used in the case of describing a non-living thing, say, a portrait?
I go for singularity (or singular), meaning
The quality or condition of being singular.
A trait marking one as distinct from others; a peculiarity.
Something uncommon or unusual.
The French language has a phrase which summarizes the notion of singularity, and that is
je ne sais quoi,
which is defined as "an intangible quality that makes something distinctive or attractive."
In English we might say about something with singularity (such as a portrait),
It has a certain something. I can't quite put my finger on it, but it's there just the same.
We know personality is too much connected with people, maybe because it has the word "person" in it or maybe something else, so what we need is a word that means the same and can be used for things other than an actual person. I use Individuality because both a living or non-living thing can have individuality.
distinctive or unique character or personality
the qualities that distinguish one person or thing from another
the state or quality of being a separate entity
All three of the above definitions impart separate shades of traits to the word. When used properly it can get the intent through.
Example: A painting of great individuality