That is, given Lead actor/actress VS Supporting actor/actress, what were the studio's reasons? (Or maybe there was some Academy rule involved?)
According to Wikipedia, a supporting actor/actress performs a role in a play or a film that "supports" the lead actor/actress.
These range from minor roles to principal players and are often pivotal or vital to the story as in a best friend, love interest, sidekick (such as Robin in the Batman series), or antagonist (such as the villain). They are sometimes but not necessarily character roles.
The film was titled My Cousin Vinny. The plot of the film focused on Vinny with Marisa Tomei's character, Mona Lisa Vito portraying his love interest.
It is a process that is decided by the voting members of the Academy, according to this article:
Ultimately, it's up to voting members of the branch of the Academy to determine if an actor or actress has a lead or supporting role while they are casting their votes, which is why studios try to influence the voting beforehand with the campaigns.
The studio behind the movie does mount their own campaign for a particular actor or actress in either a lead or supporting role.
There has been controversy with other actors and actresses in regards to what constitutes a lead or supporting role.
Timing on screen isn't everything when voters cast their ballots. For example, Anthony Hopkins won the Academy Award for Best Actor in The Silence of the Lambs (1991), yet his character was only on screen for about fifteen minutes of the film.
Rooney Mara was nominated for Best Supporting Actress for 2015's Carol, though she had a comparable amount of screentime to Cate Blanchett, who was nominated for Best Actress for the same film. Critics argued that The Weinstein Company, which launched campaigns for the actresses, made the distinction because it did not want Blanchett and Mara competing with each other in the same category.
There was also controversy surrounding Marisa Tomei's Oscar win, which unfortunately overshadowed her terrific performance and well-deserved Best Supporting Oscar win.