Similar opening positions and structures When I first started chess, my mentor asked me to learn one opening that I like and stick to it, study it and master it. But things will not always go the way you think, your opponent might not always start 1. e4. However, starting 1. e4, 1. d4 , 1. Nf3... might leads to the same position after a sequence of moves (considering the in between moves to reach that position does not consist of a blunder). 

What is the most common opening position(s) / structure(s) to be in regardless of the starting move? This is because by studying those, I think I can try to shape the board into those positions which I'm comfortable in.
There are two paths you can choose and they both depend on the current pawn structure you play.

You see, I understand exactly what you face here, as I have the same problem. When you work or have other reasons not to devote yourself to studying chess, you need to learn the least amount of openings possible in order to focus on more important stuff such as tactics/endgames/middlegame.

Therefore you need openings that are immune to transpositional tricks, but on the same side you can reach their middlegame positions as Black anytime. At best, you will need two openings -> against 1.e4 and against everything else.

As I have said, there are two choices:

FIRST CHOICE :

CARO-SLAV PAWN STRUCTURE

You can use Caro-Kann defense against 1.e4 and Queen's Gambit against 1.d4. These two openings have a lot of similar middlegames and studying pawn structures for both sides will greatly help you. Compare the below diagrams to see what I mean:

[Title "Queen's Gambit"]
[fen "8/pp3ppp/2p1p3/3p4/2PP4/4P3/PP3PPP/8 w - - 0 1"]


And this one:

[Title "Caro-Kann"]
[fen "8/pp3ppp/2p1p3/3p4/3PP3/8/PPP2PPP/8 w - - 0 1"]


Now consider exchange variations that can crop up from those structures:

[Title "Queen's Gambit declined"]
[fen "8/pp3ppp/2p5/3p4/3P4/4P3/PP3PPP/8 w - - 0 1"]


And notice that this is the same position only with the colors reversed:

[Title "Caro-Kann"]
[fen "8/pp3ppp/4p3/3p4/3P4/8/PPP2PPP/8 w - - 0 1"]


This means that if you take time to learn Carlsbad pawn structure you could cut down the amount of work drastically.

Furthermore, both Caro-Kann and Queen's Gambit declined are immune to transpositional tricks so White is unable to "kick you out" of your opening choice:

[Title "Caro-Kann"]
[fen ""]

1.e4 c6 2.c4 d5!


This is a line from Caro-Kann and is considered unclear. On other moves Black simply transposes into well explored lines as well.

[Title "Queen's Gambit"]
[fen ""]

1.c4 c6 2.e4 d5!


And you are back into Caro-Kann. On other moves Black simply transposes into well known lines.

On 1.e4 you could also use French defense, but it will require more work on middlegame structures, as they differ from the ones in Queen's Gambit. Still, the total number of lines you need to learn is lesser than the ones in Caro-Kann :

[Title "French defense"]
[fen ""]

1.e4 e6 2.c4 d5!


This is a good line for Black, covered in any repertoire book for French defense. Also, if White tries to trick you to not play Queen's Gambit he will fail and you can transpose into French defense:

[Title "Queen's Gambit"]
[fen ""]

1.c4 e6! 2.e4 ( 2.d4 d5! ) 2...d5!


And in both cases you are "in your opening", thus refuting White's try for tricking you out.

SECOND CHOICE:

The second choice is Sicilian defense-Dragon variation or Modern defense for 1.e4 and King's Indian defense on 1.d4.

A lot of work is required here, but these openings give better winning chances for Black than Queen's Gambit and French / Caro-Kann defense. They are also immune to transpositional tricks.

The pawn structure is nearly identical for all of these openings:

[Title "KID, Sicilian Dragon and Modern"]
[fen "8/pp2pp1p/3p2p1/2p5/8/8/8/8 w - - 0 1"]


It all depends on what you want:

QGD + Caro-Kann/French give you positional play, yet KIA and Sicilian/modern tend to be much sharper.

I chose QGD and French, and never regretted it, because I always "stayed in the opening" and the ideas are clear and simple. Still, I must add that these are harder to play then KIA and Sicilian/Modern, and achieving victory with Black is nearly impossible against quality opposition.

In the end, I recommend you Andrew Soltis-Pawn Structure Chess to help you grasp the middlegame concepts. With proper repertoire books you should be OK.

Hopefully this helps you a bit, if you have further questions leave a comment and I will try to help you.

Best regards.