A word for 'the ability to see beauty'

Solution 1:

I think the best is

aesthete
a person who is appreciative of and sensitive to art and beauty.
-- Google Dictionary

So the trait is "aestheticism".

^{Found with the help of onelook.com, with their the reverse search, by searching "one who sees beauty in everyday objects"}

Solution 2:

Simply, eye has this meaning.

Though, it is mainly used in the idiom to have an eye for something.

the faculty of intellectual or aesthetic perception or appreciation

an eye for beauty

[merriam-webster]

As a general word, you can use appreciation.

Awareness or delicate perception, especially of aesthetic qualities or values.

[thefreedictionary]