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.