It appears that in the Fullmetal Alchemist universe, those who are state alchemists are not necessarily in the military hierarchy per se, even if they have a rank equivalent to major.
Yes, there are alchemists such as Mustang, Basque Gran, Armstrong, etc. who are active members of the military, but there are also other state alchemists who, while involved in the military state alchemist program, are not involved in the military full-time as those I mentioned earlier are.
An example of this would be Shou Tucker, who we never see wearing a uniform, but is a state alchemist. Given that Edward similarly seems to not be involved in the military as a soldier on a daily basis, it makes sense that he doesn't wear a uniform.