How can this be right?

I'm a beginner, but the engine tells me that I am playing the Colle system, but I thought I was playing a Caro-Kann defence. I'm sure I'm being a bit daft, but any suggestions on what I'm missing most welcome. 

Opening classifications by software are notoriously silly. Sure you can trust them to identify major openings when they come from main lines, but transpositions and obscure lines are often just nonsense.

This is a reasonable example. First of all the colle system is something white does. So while it may be true that you've played the Colle System 37 times, that's fairly irrelevant to this game where you're black.

Additionally, this opening hasn't taken on a character yet. It may indeed become a colle, but for example if this game continues 4.c4 c6 then it's transposed into a semi slav like position. In general this game may still transpose into various members of the queen's gambit family where white has chosen to put a pawn on e3 before moving the bishop out. Or, if white choses to put the pawn on c3 instead, it will be a colle.

Anyway, the Caro Kann is a defense black chooses after white plays 1.e4 and black plays 1...c6. After white played e3 in this game the chance that it would transpose into a caro was essentially zero.

To make it less confusing for you... strong players more or less identify an opening by the pawn structure.

So for example any position where white has a pawn on e4, is missing the d pawn, and black is missing the c pawn, will be some kind of sicilian.

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And now black's pawns will define the type of sicilian. a6 is a Najdorf, g6 is a dragon, pawns on d6 and e6 is a scheveningen, a pawn on e5 is a sveshnikov.

Most openings aren't quite that pawn heavy, the Sicilian just makes a convenient example.

The structure below is from the super classical Ruy Lopez or Spanish

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And now instead of pawns, the way black chooses to develop the knights and bishops will define which variation it is... but the structure is what makes it a Ruy / Spanish.

In your position, I'd wait to see what both players are doing with their c pawns before trying to classify it... and that's a problem with software. Sometimes it classifies something too early and when the structure takes an unexpected turn it doesn't adjust the name. If you asked a grandmaster what the opening of your game was after 3 moves, they wouldn't give you a name yet.

It's a combination of pawn structure and piece placement.

For example, the Colle, London, and Torre, the pawn structure is the same.  The difference is that the Torre sees the Bishop on g5, the London sees it on f4, and in the Colle, it's on c1 or d2 (behind the pawn chain).

However, some openings are mis-identified, even at the human level.  I see many call an opening "The Jobava London".  There is no Jobava London.  There is the Jobava Attack, and there is the London System.  They are NOT the same thing.  The White Queen's Knight is the difference between the two.  If it goes to c3, you are in a Jobava Attack, while if it goes to d2, you are in a London System.

It's a lot like the difference between a Kan and a Taimanov Sicilian.  Just because you play 4...a6 does not make it a Kan.  It all has to do with the Queen's Knight.  If it goes to c6, you are in a Taimanov.  If it goes to d7, you are in a Kan.

For example, 1.e4 c5 2.Nf3 e6 3.d4 cxd4 4.Nxd4 a6 5.Nc3 Nc6 is a Taimanov, not a Kan, despite the 4...a6 move.  This position can also arise from 4...Nc6 5.Nc3 a6.  5...a6 is the older line, the more modern approach is 5...Qc7.

So yes, openings are not identified by moves.  They are identified by pawn structure AND pieces.  In fact, attempts at a Colle could easily reach a Caro-Kann.  For example, 1.d4 d5 2.Nf3 c5 3.e3 cxd4 4.exd4 is a Caro-Kann Defense!  1.e4 c6 2.d4 d5 3.exd5 cxd5 4.Nf3 is the exact same position!

Sure, development + pawns (so the whole position) is what defines an opening. A classic QGD vs a Nimzo vs a Ragozin, slightly different pawn / piece choice, but very different openings.

I still tend to think of the pawns as defining the family (so to speak) with piece placement narrowing it down after that... but maybe that's an oversimplification.

A Caro-Kann Defence requires c7-c6 to be played which hasn't been played yet

The Caro-Kann has c6 as their first move for black. 

Actually, you can reach a Caro-Kann without ...c6 by Black, but not the way the OP has it.

1.d4 c5 2.e3 cxd4 4.exd4 d5 or 1.d4 d5 2.Nf3 c5 3.e3 cxd4 4.exd4 are both Caro-Kanns, yet Black never played ...c6.

Caro-Kann is this:

In your example your opponent didn't play 1.e4 (so you didn't get the opportunity to play 1. ...c6 after e4), so this is not Caro - Kann. There are certain openings that can transpose to Caro-Kann defense as well with different order of moves but this opening white employed is far from it.

Yeah, it transposes to exchange variation position because c pawn is not there anymore, even though it didn't start with c6.

@OP

Anyway, Caro-Kann is generally 1. e4 c6, that is all you need to know for now, and what you posted is very different.

Thanks everyone! I’m blown away by all the helpful comments! Realised my mistake about the Caro-Kann, silly - even I know that. But a lot of the other comments will require some time to process! But thank you all so much!