In practice I'm a little more forgiving about two creatures sharing a space than the rules strictly allow; my usual exception is for one standing and one prone beneath them. I will also typically allow a creature to move into the space of an invisible foe, assuming it isn't an especially large/bulky/immobile invisible creature; as part of that I assume that the invisible person ducks to the side a bit to not give away their presence.
But as others have noted, moving into the same space as another creature and attacking from there fails for me on two counts:
1. It creates really weird and unnatural game states, where (for example), a big plate-armored fighter can stand in and block a doorway and have his whole party sequentially move into his space, attack, and move back again in the space of six seconds. That seems exploitative. It'll seem even weirder to the players if I have a full dozen or more enemies do it.
2. It breaks my suspension of disbelief. I know how much space I need to swing a sword. Having a person literally shoulder to shoulder and impinging on the space for my arms to swing will make it harder for me to attack.
To be fair, with certain weapons and training, full 5' squares per combatant would definitely not be needed. If me and my shield buddy were fighting in shieldwall with short, stabbing swords or spears, we'd probably only need 3' of frontage or less each. OTOH, with a claymore or great axe, I'd need a great deal more than 5' total frontage. 5' per combatant is a nice simple rule to keep it easy.