Is there any reason why he's using "feel" instead of "wish" or "hope"?
In this case, "feel" has meaning close to "believe," "wish," and "hope." However, he used "felt" to show that his birth mother held strong feelings towards (or a strong opinion about) something, hence "felt." You can use this expression in similar situations. Ultimately, any of these words can be used to express the same general meaning. Ex: I feel very strongly about this topic. Ex: My mother felt very strongly that I should study abroad.