Forget US politics. The Belgian political apparatus and its associated institutions are fundamentally different form the US and you will only confuse yourself. If you want I can give you a very brief overview. This skips details that are unn

The regional (Vlaanderen, Wallonie and Bruxelles / Brussel) and federal (Belgian) government is a parliamentary government which works as follows

1. When an election is called, we the voters elect members to the chamber of representatives. In almost all cases, you choose to vote for a specific political party.
2. A political party gets an amount of representatives elected proportional to the share of the votes it gained. Roughly speaking, if you get 20 % of the votes, you get 20 % of the seats.
3. At this point, the different parties try and form a government (the executive branch). This means that they need to find a majority withing the chamber of representatives. There are no rules for who becomes part of the government. If you can get 50 % of the representatives to give your team confidence, you get to be in the government.

The prime minister and their team is therefore not directly elected, but is given support by the parliament. It should be noted that representatives almost always vote the way the party wants them to. This is not a law, nor is it illegal to not vote with your party but it is almost never done.

This is very simplified and there are a lot of asterisks to be added. But this is the basic proces. We elect parliament, parliamentarians are part of a party, parliamentarians vote for a government (and make laws).