How to design a sports statistics database for multiple sports?

The proper way is to make multiple tables, one for each sport seems likely.

Player (Player_ID, FirstName, LastName)
Team (Team_ID, Sport_ID, TeamName)
TeamList (Team_ID, Player_ID)
Sport (Sport_ID, SportName)
HockeyStats (Player_ID, Team_ID, Year, Goals, GamesPlayed, Assists)
BaseBallStats (Player_ID, Team_ID, Years, BoringSport)

This also resolves the situation with trading, which team the points were obtained from, as well as multiple sports.

You can get inspiration from sportsdb, a rdbms schema for sports modeling created by XML Team. In that schema the game and players statistics are stored in denormalized tables, and the core stats (like score) are kept separated from the sports-specific ones. The schema is quite complex, but can give you an idea about a possible implementation

An alternative approach is a single table with a statistic-ID column, identifying which statistic is being described in that row. This may work well if the statistics to be kept will change over time. It will only work well if the basic form of each statistic is the same (sorry, I'm British and not familiar with the sports and stats you mention), but there are multi-table minor variants (all percentage stats in one table, all simple-count stats in another, etc).