No, the Electoral College is a special system that only applies to US Presidential elections.
In the United States, election rules are determined by individual states in accordance with their own Constitutions. For state-wide offices like the governor, states have generally (though not always, see below!) settled on simple first-past-the-post majority vote systems.
US Presidential elections work differently because Article 2, Section 1 of the Constitution sets out specific rules for the election of the President. Since the electoral college is laid out in the Constitution, it can only be changed via a constitutional amendment, which is a rare and challenging procedure.
Interestingly, in the Mississippi governor election, there actually is a system that functions similarly to the electoral college. To win the governor's office, a candidate does need to win a majority of house district (or a majority in the state House):
The winner of Mississippi's gubernatorial election Tuesday will not only have to capture the state's popular vote, but will also have to prevail in the state's unique election process for electing a governor and other statewide officials that was established during the Jim Crow era
A candidate needs a majority in the popular vote and needs to win a majority of Mississippi's 122 state house districts. If no candidate fulfills both of these requirements, the Mississippi House of Representatives, which is controlled by Republicans, selects the winner.
The election process, as written in the state's constitution in 1890, was enacted at a time when white Southerners were putting in place laws to deny blacks political power.
How Mississippi's Jim Crow-era election system could decide Tuesday's election - CNN
Update: This is no longer the case. In the 2020 election, Mississippi voters passed an amendment to the State Constitution that removed this system. Now, the governor and other statewide officials are elected by popular vote, as in all other states.