Everyone is proficient with their own unarmed strikes
In the errata for the Player's Handbook, the Unarmed Strike entry was removed from the weapon table. However, the rules on Unarmed Strikes in the combat section now say that every character is proficient with unarmed strikes (emphasis added):
Instead of using a weapon to make a melee weapon attack, you can use an unarmed
strike: a punch, kick, head-butt, or similar forceful blow (none of which count
as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 +
your Strength modifier. You are proficient with your unarmed strikes.
Dale M's answer is pretty much the one I would have given, but I'll try to add clarification based on the comments from his answer.
On a single attack, a wizard and a fighter of equal stats and features have similar chances to hit. But it doesn't matter because a fighter will still be a better fighter than a wizard will be. It also raises the point that the wizard shouldn't have the same stats and features after level 1.
This is related to the change in philosophy of 5E: the number in the attack doesn't grow that much, but more interesting tools and tricks make the difference between a good fighter and wannabe-fighter. Example include sneak attack, battlemaster's maneuver, bonus action extra attack, conditional extra damage (ex. the ranger's colossus slayer).
How is a fighter's training shown in the mechanics?
A fighter's training doesn't appear as bigger to-hit number. It appears as extra attack, more HP, fighting style (the fighter goes beyond blindly hitting things), more reliable crits, combat related feats, more efficient offhand attacks...
Bonus attacks, fighting style, feats, stat boosts are not new to 5E, but they still make a difference when it comes to actually hitting. They actually make more of a difference because the base numbers are closer. And then add more damage once an attack does hit.
I will ask the reverse question
If a wizard and fighter have equal experience in fighting (aka they are level 1 and have never seen a real fight), why should the fighter have more chance to hit? They only ever had basic training experience. Or at least even the fighter may have never seen a real combat. While the wizard may at least know enough his staff to hit a target.
Best Answer
Training proficiencies don't stack.
It means 4+2 (ability and proficiency) as your ability is 18 for a rapier, or a hand crossbow, or a long bow, or a short sword. There are no other bonuses unless you left some out in your question.
The overlap between Drow and Ranger proficiency in weapons does not add / stack proficiency.
If you had made a Drow wizard, the wizard would have the same race based weapon proficiencies -- short swords, rapiers, hand crossbows -- which is a good deal since a normal wizard does not.