The feat doesn't limit the amount of attacks you can make. It only states that whenever you make a melee attack against a creature, you don't provoke opportunity attacks from that creature for the rest of the turn. In fact, multiple attacks work quite well with this: if you are adjacent to two enemies, you can make one attack against each and then leave their reach without provoking an opportunity attack from either!
Can I make more attacks if I have Extra Attack, or am I only allowed to take one Attack action?
In most cases, even if you have Extra Attack, you will only take at most one Attack action during your turn. Extra Attack doesn't grant you additional Attack actions, but increases the amount of attacks you can make whenever you take said action. Note that Mobile feat's bonus triggers off any melee attack, including melee spell attacks, opportunity attacks and other attacks made without using the Attack action.
The distinction between the Attack action and an attack as a game mechanic is notable as there are features that are triggered by taking the Attack action as well as features that are triggered by making attacks --- see this question and its answers for more information about the distinction.