If we say that one quantity is directly proportional to another quantity, does this mean that they are equal to each other?

It doesn't mean it equal to each other. lets say x is directly proportional to y. Here, if something happens to x then, y will do the same thing. for example, if x increases, then y also increases. similarly, if x decreases, then y also decreases. therefore, x is something or lets say it is a major factor that decides the fate of y.

NO!

It means that (one of them) divided by (the other one) is always the same number.

(Other people probably won't explain it to you this way, but I'm special, and this is true.

Anyway, the direct answer to your question is: No.)