Why does user column name has UNIQUE Constraint? I got aware that the column name in TABLE users has a UNIQUE Constraint, when I tried to register a 2nd user with the same username : it returned a `integrity constraint violation 1062 duplicate entry '' for key 'name'.

Lets' say 2 of my customers called "John Doe" want to create an account : Ths 2nd one could not then ?!

So I made a quick SQL fix 

ALTER TABLE `users` DROP INDEX `name`


I'd like to be sure it's not a very bad idea for some reasons ?

Config : this is a Drupal Commons distro
It's a bad idea, your are modifying a core feature of Drupal that others module (and the core itself) may relay on. That's the reason why that column has 'UNIQUE' option, because Drupal requires unique username. Think on user login: how you can login with different users if the have the same username?

Check this answer where you may find more info:
Is it possible to allow duplicate usernames ?