It's certainly possible to quantify subjective criteria
People do it all the time. I mean, they rate potato chips on "mouth feel", right? So you can come up with some criteria. See the other answers.
And if that's what you really want, go for it
You're going to get some side-eyes
At a guess, I think you're going to get some side-eyes with this question, because on the surface it seems a little silly, "can't be done, and why would you want to, go on and just play the game, will you?" seems the obvious answer.
Don't worry about it, fun is fun
But don't worry about that. If you're having fun, you're having fun. There are a lot of ways to play D&D. People spend countless hours making adventures, maps, building PCs an NPCs, reading lore. For some strange reason I like reading questions on this board and spending sometimes an absurd amount of time researching and writing answers. I think someone who does that has no room to judge how someone else plays D&D.
A frame challenge
Thanks to Eddymage for suggesting this is actually a frame challenge, and to Novak for his comments about monster CR.
You can quantify PC's like potato chips. You might even spend a month and come up with a PC rating system as broken as monster CR.
And if that's fun for you, go for it.
But PCs are Player Characters because they are characters (like from a story) that players make. Monsters have a Challenge Rating because they are "challenges", as in challenges for the players to overcome.
Monsters have a CR because they're just stock images for the DM to build challenges with.
Read the introduction to the Player's Handbook. It's full of phrases like "The Dungeons & Dragons roleplaying game is about storytelling in worlds of swords and sorcery".
Sure, D&D is a big tent, and plenty of people only ever play their characters as interchangeable sets of numbers "fighting" other interchangeable sets of numbers and if that's fun, then that is fantastic.
The first chapter of the PHB says "Your character is a combination of game statistics, roleplaying hooks, and your imagination." The statistics are only part of the character. An important part, to be sure, but just one part.
There is room in D&D for characters to transcend their stats.
I'm not sure rating even "fun" or "satisfying" or "fulfilling" on a scale of 1 to 5 is enough for such characters.
A story
I one time joined a campaign at 9th level with an arcane trickster. It was a pretty good build. I vaguely based the personality in part on a character in a book I read many years ago, and there was one scene when the character rode a donkey as part of a scam, pretending to be a kind of wandering fool. I thought it would be a cool way to introduce the character to the party. I asked the DM if I could have a donkey. They shrugged, and said, "sure".
I didn't realize I'd be meeting the party in the middle of a dungeon, but that's where I joined them, this guy leading an extremely reluctant donkey -- "excuse me, do you know the way out of here", kind of thing. It was great. That donkey was never a mechanical advantage, not once. When the monsters showed up, the donkey just somehow had a knack for not being there. It couldn't (or wouldn't) carry any more than the character could. I would even cast fly on the donkey sometimes, but mechanically it was the same as casting fly on myself. By any measure of party power that donkey just didn't even exist.
I ended up developing a lot of depth to that character. I had conceived of him as a stone-cold killer with a heart of gold. Over time, I developed his backstory in depth. I initially thought the donkey would just be a fun way to meet the rest of the party, but the story of the donkey grew and grew, the importance of the connection between the rogue and the donkey grew and grew, and the donkey even got its own backstory. Yeah, it was a little silly, but it was a very satisfying part of that campaign. I actually ended up switching to a different character, and the rogue and the donkey continued their story off-stage. We finished that campaign at level 20, and retired the setting. But somewhere in the lands of imagination there is a band of adventurers, now retired, whose lives and deeds became the stuff of legends. Among their number is a stone-cold killer with a heart of gold and his life-long donkey companion, who now nibbles the best parsnips that a very great deal of money can buy, in a very, very, very nice stable. Occasionally they can be seen high up in the sky, against the sunset, patrolling a now prosperous but once devastated and dangerous land.
Power? It really doesn't matter, but let's say 4 out of 5, I mean he was a really great arcane trickster.
Awesomeness? Off the charts.
No power rating is going to find that awesomeness. You find awesomeness by playing.