私は彼女の考え方について興味があります。
Although this sentence is usually translated as "I'm interested in ~" with "I" as the subject, the grammatical subject of the original sentence is not 私. A very literal translation of this sentence is:
As for me, interest exists regarding her way of thinking.
As you can see, the subject of あります is 興味, not 私 nor 考え方. Of course 興味 is an inanimate object. In other words, this sentence is about the existence of someone's interest, not about the existence of someone ("me" = 私). Since this sentence is not about where I am or whether I exist, you cannot use いる here.
Japanese is a topic-prominent language, and one sentence can have both a は-marked topic and a が-marked subject. You may have learned this fact using a sentence like 彼は背が高い ("He is tall") or ゾウは鼻が長い ("An elephant has a long nose"). Please keep in mind that the subject is 背 and 鼻, respectively, no matter how this type of sentence is usually translated into English.
Really confusing to me. Someone has explained it well below though, so i'll digest it. thanks.
– sups12 Jul 26 '20 at 00:01