"We can think of an abstract noun as being similar to an abstract painting. Both abstract nouns and abstract art represent ideas instead of concrete objects."
Why do we need "being" in the original?
I hope I have parsed it correctly:
We : subject
Can think of : modal plus main verb plus preposition
" Of "preposition and "an abstract noun"- an object of preposition
"as" a preposition with "being similar to an abstract painting
Is it just one clause - the whole idea or made up of two sentences?
We can [ think of an abstract noun as [ being similar to an abstract painting ]] . The "being" clause is the comparative equivalent to saying "An abstract noun is similar to an abstract painting". The sentence consists of a matrix clause (the sentence as a whole) with two embedded subordinate clauses, the one in the outer brackets a complement of "can" in clause structure, the one in inner brackets a complement of the preposition "as" in PP structure.
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We can [think of an abstract noun as [being similar to an abstract painting