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I am using Lichess's master database and after 1. d4 Nf6, 71.63% of the game continues with 2. c4. However, after 1. c4 Nf6, only 5.22% of the game continues with 2. d4. Are they not essentially the same opening?
My question is:
Why after 1. d4 Nf6, 2. c4 is so common but after 1. c4 Nf6, 2. d4 is so rare? Is it more because of some psychological reasons, or is it because after 1. c4 Nf6, moves such as 2. Nc3, 2. g3 and 2. Nf3 are objectively better than 2. d4?
To answer your question, I think it mainly comes down to selection bias. The White players of the games after 1.c4 Nf6 are often not intending to play a Queen's pawn opening (at least, not right away with 2.d4). I'll go into why in the rest of this answer.
If you wanted to play the position after 1.d4 Nf6 2.c4, there's (almost) no good reason to not start with 1.d4. Starting with 1.c4 gives Black the opportunity to play 1...e5 or 1...c5 (both highly respectable systems), and now you're stuck in an English.
After 1.d4, the main downside I can think of is that it allows the QGA (1.d4 d5 2.c4 dxc4). I think it'd be fairly odd to play 1.c4 just to avoid the QGA, but I'm sure there are some players out there who do this.
Note that the 1.c4, 2.d4 move order does not have the benefit of avoiding the QGD. After 1.c4 Nf6 2.d4, 2...e6 gives Black the option to follow up with 3...d5, should he choose to do so. Black could even meet 1.c4 with 1...e6 if he wanted to.
For the Slav, there's mostly nothing much the 1.c4 move order does. I say mostly because after 1.c4 Nf6 2.d4 c6, 3.Bf4!? is a possibility. Now if 3...d5, White has time to play 4.e3. This reaches a position where the dark-squared bishop is developed, and White is in time to prevent Black from winning a pawn with ...dxc4. However, Black can instead just meet 3.Bf4 with 3...Qb6, and the game should be equal.
Black could also meet 1.c4 with 1...c6 (intending ...d5 immediately on the next move), although here he'd have to be fine with a possible transposition into the Caro-Kann Panov after 2.e4.
See Cephalopod11's answer for more good info on this topic.
Looking at this through the intentions of the moves helps me make sense of this.
The move 1. d4 establishes a dark-square pawn in the center. Light-square control however is still missing. If possible, White will remedy this with an immediate 2. e4 but one must wait for Black's reply.
After 1. d4 d6 there's no immediate challenge and White can take the central light-squares with 2. e4:
[FEN "rnbqkbnr/ppp1pppp/3p4/8/3PP3/8/PPP2PPP/RNBQKBNR w KQkq - 0 2"]
This controls both the light-squares and the dark-squares and Black must attack the center. If Black does not fight-back then Black will be disadvantaged and may even lose the game.
1. ... d5 is a fine response. It takes the central light-squares but there is an interesting downside. Somehow White's dark-square control has increased too - Black's pawn on d5 can never retreat to d6 and support a dark-square break:
[FEN "rnbqkbnr/ppp1pppp/8/3p4/3P4/8/PPP1PPPP/RNBQKBNR w KQkq - 0 2"]
In fact, White has the dark-squares all wrapped up suddenly, why not challenge the light-squares with a Queen's Gambit?
[FEN "rnbqkbnr/ppp1pppp/8/3p4/2PP4/8/PP2PPPP/RNBQKBNR w KQkq - 0 2"]
So 1. ... d5 works, but 1. ... Nf6 is a different try. Note that this move also controls the light-squares in the center. It also doesn't commit the d-pawn. I am not a master but I feel like this is why 1. ... Nf6 is said to be more flexible.
[FEN "rnbqkb1r/pppppppp/5n2/8/3P4/8/PPP1PPPP/RNBQKBNR w KQkq - 1 2"]
After 1. ... Nf6 White still wants to at least make a try for the light-squares. And why not? Just like in the Queen's Gambit, we have the dark-squares wrapped up, let's get the light-squares too.
The trouble is that the Knight controls e4 - well, no problem, we know we can play 2. c4:
[FEN "rnbqkb1r/pppppppp/5n2/8/2PP4/8/PP2PPPP/RNBQKBNR w KQkq - 0 2"]
Game on - and as you say, there's nothing wrong with getting to this position via the 1. c4 Nf6 2.d4 move-order. But remember our point about 1. ... Nf6 - that it was more flexible? Well, so too are there some down-sides to 2. d4. Notice, for example, that the dark-squares aren't under-threat. Maybe we could get that dark-square control by developing a piece:
[FEN "rnbqkb1r/pppppppp/5n2/8/2P5/5N2/PP1PPPPP/RNBQKB1R w KQkq - 2 2"]
This also reserves the right to play d3 later, possibly supporting the e4 square. Imagine this:
[FEN "r1bqkb1r/pppp1ppp/2n1pn2/8/2P1P3/3P1N2/PP3PPP/RNBQKB1R w KQkq - 0 4"]
Or you could go full hyper-modern:
[FEN "rnbqkb1r/ppp2ppp/4pn2/3p4/2P5/1P3N2/PB1PPPPP/RN1QKB1R w KQkq - 1 4"]
2. d4 doesn't contribute to these plans and it's definitely not urgent, so why not try something else?
The first moves 1.d4, 1.c4. and 1.Nf3 can all potentially transpose to the same opening lines, but they can also each avoid specific openings, while also allowing others:
1.d4 gives Black the full range of both solid, well respected defenses, as well as some of the trickier, less sound responses:
1.c4 allows Black to play 1...c5 and 1...e5 (entering lines that are not possible via the 1.d4 move order), but also give White some ways to avoid some of the best responses to 1.d4:
Many 1.c4 players are either intentionally trying to avoid something like the Nimzo, or they just like the Symmetrical and King's English (1...c5 and 1...e5) and are hoping that's what the opponent will play.
Of course, some players just like to throw off their opponent, so they start with the less common 1.c4, but if they are actually happy facing all the Indian Defenses (1.d4 Nf6), usually they would just start with 1.d4.
