Using the feat in this way does nothing. You cannot add your proficiency bonus again.
The rules are quite specific. From the Players' Basic Rules, under "Proficiency Bonus":
Your proficiency bonus can’t be added to a single die roll or other number more than once. For example, if two different rules say you can add your proficiency bonus to a Wisdom saving throw, you nevertheless add the bonus only once when you make the save.
Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue’s Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.
So no, if you're proficient in martial weapons and you take the Weapon Master feat and choose longswords, you don't get any more proficient. The feat is for characters who aren't already proficient.
As an edge case: notice that Fighters and other classes are proficient in "simple and martial weapons" but "all armor and shields" - the wording leaves open the possibility that simple and martial weapons might not cover all of them, and indeed it doesn't cover improvised weapons (though if that's what you're after, Tavern Brawler is probably a better choice.) If you come across an exotic weapon (though Wizards hasn't published any so far), this feat might be useful for these classes as well.