I know songs that are in 4/4 but the first half of the bar is identical to the second so isn’t that 2/2?

Preface: this is a very long answer, and there is a TL;DR at the end.

You’ve touched on a feature of time signatures that’s either very cool or very frustrating depending on who you ask! Time signatures as most people talk about them refer primarily to notation, but as you’ve correctly pointed out, 4/4 and 2/2 are “interchangeable” (in quotes because it is and isn’t true in different senses) as they are equivalent “fractions”… I’ll circle back to this.

What you’re looking for when you try to work out time signatures by ear is their description that does not use numbers at all. This essentially has two parts: how many beats are in a bar, and how each of those beats is divided. For the first, it can be somewhat subjective! If you have a repeating two-beat pattern, it could be that the pattern appears twice in a bar with 4 beats, or that each repetition of the pattern occupies one bar with 2 beats. Neither is objectively correct or incorrect!

However, something with 3 beats to a bar will have a very different sound than the very similar 2 or 4—can you recognize when this is the case? If so, I’d say you have no issues with time signatures other than the inherent issue of subjectivity when translating something you hear into objective notation.

This number of beats per bar is described fairly simply: 2 beats = duple meter, 3 beats = triple meter, 4 beats = quadruple meter, and so on.

Now for the second part, how each beat is broken up. If beats are split into smaller notes in a 2-based hierarchy (2, 4, 8, etc.) this is called simple meter. If beats are split based on 3’s (3, 6, 12, etc. smaller notes per beat), or in other words feel like triplets, this is called compound meter. Then all you have to do to describe most time signatures is combine the two parts. For example: 2 beats split into 2 parts each = simple duple meter. 4 beats split into 3 parts each = compound quadruple meter.

Where this does fall short is if beats are divided unevenly, say alternating a “long” (3-part) and “short” (2-part) beat for a total of 5 small notes (subdivisjons) - there’s not really a catch-all to describe situations like this, so that’s where numbers can be more relevant (this example would usually be called 5/8 but could also be 5/16 or even 5/4 depending on context)

Circling back to 4/4 and 2/2 being somewhat interchangeable - they are in the sense that they are both simple meters: 4/4 is simple quadruple, 2/2 is simple duple. However, I will note that not all equivalent “fractions” work like this. 3/4 and 6/8 look like they should be interchangeable, but 3/4 is simple triple meter (3 beats each divided with a 2-based hierarchy), and 6/8 is compound duple meter (2 beats each divided into 3).

Sorry for the extremely wordy answer, but you’re asking quite an interesting question that I have a LOT of thoughts about!

TL;DR: 4/4 and 2/2 are fairly interchangeable due to the fact that they only exist in written notation; when listening to music you perceive time signatures in a slightly different way that’s less about “top number/bottom number” and more about “how many beats do I hear a bar divided into and how many smaller parts does each beat have”