Mathematically speaking, the 9 doesn't contribute any meaningful information. The same information would be encoded in the ERA using Earned Runs/Innings Pitched
ERA stands for Earned Run Average and to put it in other words, the average number of runs given up per game. Earned runs divided by innings pitched gives you the average number of runs per inning, which isn't particularly meaningful or easily understood. Multiple by 9, though, and now it becomes easy to grasp.
For example, at the start of the season Pitcher A pitches a complete 9 inning game win and gives up 2 runs in the game. ERA is (2/9)x9 which gives you 2. One complete game, two runs, and the ERA makes sense and is easy to understand. If you report ERA as .222 (or 22.2) that doesn't make any sense to how that works out in terms of evaluating pitcher effectiveness overall.