Literary critics would argue that all writers belong to all schools; the real key is what you look for in a novel. Thus, both a historical critic and a postcolonial critic might be interested by the relationships between colonial India and Great Britain as described in a "Passage to India"; a Marxist critic might be interested in the relationships and how power is gained and lost throughout the novel.
Perhaps you are wondering what literary era Forster belonged to? This is an interesting question because he wrote during the time of the modernists, but his interests in class differences and tightly-structured narratives had a lot more in common with Victorian writers than it did with the disjointed, stream of conscious style of his age.
You ask an interesting question because Forster goes to great lengths to avoid being associated with any school of criticism existing at the time he delivered his lectures that are the basis for this book. Instead, he examined literature based on various components such as "The Story," "Characters," "Plot," etc. According to eNotes, "Forster asserts that the most important measure by which literature ought to be judged is that of the 'human heart,' concluding that the most important ‘test’ of a novel is ‘‘our affection for it.’’ Based on this comment, if I had to categorize Forster, I would say he seems to lean toward reader-response criticism.
This type of criticism contends that the "meaning" of a work isn't merely something the writer inserted but rather an interpretation created or produced by the reader as well as the writer. For more information about this school of literary criticism, see the link below.