In Reb Chaim Soloveitchik's Wikipedia page, it says about his derech that
That focuses on precise definition/s and categorization/s of Jewish law as commanded in the Torah with particular emphasis on the legal writings of Maimonides.
Why does the Brisker Derech focus on the Rambam so much as opposed to any other Rishon?
In his article "'What' Hath Brisk Wrought? The Brisker Derekh Revisited," R. Moshe Lichtenstein argues that,
the basic change wrought by R. Hayyim was the refocusing of the learning process upon the hard halakhic data that emerge from a sugya, instead of upon the mechanisms by which they were derived. To state the point in more technical terms, the Brisker approach shifted the learner's interest from the talmudic discussion itself (shakla ve-tarya) to the practical implications thereof (nafka minahs)...This is the reason that Rambam occupies center stage in the Brisker orbit, for it is he who distilled the talmudic conclusions into pure halakhic form, systematically omitting any interpretation or mention of the accompanying discussion, presenting us only with the halakhic hard data without encumbering it with any explanations.
See there for more elaboration on this point.
Rabbi Menachem Genack makes a similar argument:
The Brisker derech involves conceptual categories. Halachic acts, thoughts and objects are all conceptualized and placed in categories. Once placed in the proper categories, the halachic and conceptual characteristics of the phenomena become more clearly defined and logically comprehensible. The Brisker emphasis on categorization explains the well-known Brisker attachment to the Rambam. The Rambam took the entire corpus of halachah and categorized all its elements in a strictly rational, analytical order. This is the same conceptual groundwork on which the Brisker derech is based.
According to the various writings shown by Google for Soloveitzchik and Rambam, the main reason is the precise, rationalistic approach of the Rambam to halacha. Rambam is the one who is known for the precise definition of the halacha. Rav Chaim Soloveitzchik therefore uses the Rambam as the prime exemplar of this method of learning.
For example, The A Priori Ideal: Aspects of Rabbi Soloveitchik’s Interpretation of Maimonides’ Philosophy goes into detail how this viewpoint applies to Rav Chaim's grandson Rav Yosef Ber (J. B.) Soloveitzchik.
