No. If someone says 'the championship game' it means a specific game in a championship competition but both speaker and listener know which game it is. e.g. I'm going to finish work early today so I can watch the championship game. This is the game in the championship that is on TV today. It's not necessarily the final.

As others have said 'championship' is usually the name of a competition. The 'final game of the season' means the last game of a league competition in football (where every team plays each other twice). However there are lot's of 'finals' for different competitions where two teams play each other to decide who wins. It gets confusing because in European football there is 'the Championship', which is the second league in England (below the Premier League) and the 'Champions League', which is the ultimate competition for the best teams in Europe. So here you would have 'The Champions League Final' and 'the final game of the Championship'.

In some sports (Basketball - NBA, Hockey - NHL), the championship series is refer to as "The Finals" because it is the last series in the playoff tournament. "The Golden State Warriors and the Boston Celtics were in the finals last season" refers to the championship series.