If you multiclass and reach levels where extra attack is given, can you stack those extra attacks? Example, a Monk 5/Fighter 5 earns Extra Attack at level 5 in each of those classes. Does that mean the character can attack 3 times (1 attack action + 1 extra attack (monk) + 1 extra attack (fighter))? Or does the character only get 2 attacks (1 attack action + 1 extra attack)?

They do not stack if they come from multiple classes, but a high level fighter can gain extra attacks that stack with the extra they already got. (4attacks per action for a level 20 pure fighter)

The multiclassing rules has this:

If you gain the Extra Attack class feature from more than one class, the features don't add together. You can't make more than two attacks with this feature unless it says you do (as the fighter's version of Extra Attack does). Similarly, the warlock's eldritch invocation Thirsting Blade doesn't give you additional attacks if you also have Extra Attack.