I think the easiest way to explain being able to see the face in the painting is that the character had seen an earlier sketch by this same person before.
Years ago, the character looked over the artist's shoulder to see them drawing something and asked what it was. The artist explained that it was an image of their face, pointing out the different contours of the painting and explaining how it looked like her. For example, seemingly random circles are the insides of her eyes. The jagged triangle shape represents a nose, and the ovals at the side represent ears.
Art is subjective, so there is no one right way to interpret it. What looks like a butterfly to one person may look like a mask to another. In the same way, while most people take a look at this and see a boat, a train, or a random assortment of shapes, our main character sees a face. If she really cares about this artist, even not knowing that it was them who made the painting, seeing the same shape might bring up old memories of that one time she saw that artist sketching out a similar shape.
It also might help if the character has a unique view of the world.
One day, the artist could have been drawing a boat, only for your main character to exclaim "Oh look, that's my face! See, those are my ears and that's my mouth."
The artist just rolls with it and laughs, redesigning it to make it look more like a face to the character.
Lastly, little details would clue her in that it is not just anyone's face, but hers in particular. For example, a series of lines have the same color as her hair. A pair of circles has the same color as her eyes. A unique shape at the bottom resembles the locket she always wears.
It's kind of like an inside joke. If you don't know the artist well, the details just seem randomly placed, but they have simple explanations only these two understand.