True ending, explaining Venom Snake's skills (HUGE SPOILER)
Solution 1:
On one of the last tape you unlock, you can hear Ocelot discussing this with Big Boss/Naked Snake. (Not the exact word since I can t listen to it right now):
"He was our best man, almost as good as you, he will be fine, and I ll be there to support him"
He is implied to be the best MSF soldier you had in Peace Walker, and like in MGSV: TPP, soldier you pick up often have equal if not superior skill to Big Boss.
The point of it is, Big Boss is nothing but a title, a name, you are just more visible, but there is hundred, thousand of men in the dark, who may or may not be better than him, but are not known to the public.
Venom just happened to be at the right place in the right time, and had a pretty extensive CV too. Best soldier of the first PMC, composed of elite soldier, and personal medic of Big Boss himself, that guy is not any no name.
You may compare the title of Boss with the Medal of Honor (or for that matter, any military reward). Your mission was difficult and is a success, you are rewarded, sometime even without having done anything heroic. And thousands died after doing something purely heroic, but nobody saw them, and they are forgotten.
Big Boss is just not the best soldier in skill. Just the one who was at the right time in the right place with the right skills, and got rewarded for it.
Example of people of similar/higher skill of Big Boss, but without title nor reward:
(As a side note, thoses also are only know to us player by the fact they were in key position, who know about all thoses no-name soldiers?)
Skullface: (Spoiler for MGS3, MGS5: GZ, MGSV: TPP):
Skullface, and a lot of XOF, sneaked Snake AND the russian for at least half of Operation Snake Eater, and supposedly helped Snake in the shadow. The reason Skullface hated Snake was he did at least as much work as Snake, without any reward.
Ocelot: (Spoiler for all MGS but PW):
Was a triple agent in operation Snake Eater, was supposedly behind the scene during PW, instructor in MGSV, and ended up being a quintuple agent for the rest of the series! (Working for Big Boss, and in this way fooled Cypher, Solidus and Liquid, while pretending to be mind controlled by Liquid). Also fooled Venom and himself in MGSV to help Phantom Snake. The guy deserve a medal bigger than Snake for that.
Solution 2:
Remember your first mission? Ocelot sends Venom in to save Miller with no backup to, "Prove to the World that you've still got it, that you still are Big Boss". Like Big Boss says in "The Man who Sold the World" -- "[They] are both Big Boss".
As to whether or not Venom's skills are actually a match for Big Boss, we won't know, but the point is, it doesn't matter -- he's good enough to convince everyone that he is Big Boss. And don't forget, he may have been a medic, but Venom was one of the (Peacewalker) Diamond Dogs himself; just because he knew medicine didn't mean he didn't know how to sneak with the rest of Big Boss' crew!
Solution 3:
He's described as one of the best soldiers from the old mother base. So when he's put into the role of Big Boss, it's not too much of a stretch that he's on par with Big Boss.
I think it's his great skill plus a bit of a placebo effect that makes him passable as the real big boss.