Let's think of the harmonics in the sound. Most musical sounds consist of a fundamental and then further overtones at (roughly) integer multiples of the frequency.
For the sake of simplicity, let's imagine that the instrument timbre we're using has 3 overtones - and that rather than C, we're playing a note of 100Hz. That means it has partials at 100, 200, 300, and 400Hz.
A fifth is a frequency ratio of 3:2, so our fifth will have its fundamental at 150Hz - and then have energy at 300, 450, and 600 Hz.
So the resultant sound has components at 100, 150, 200, 300, 400, and 600 Hz.
But now if I invert the chord...
Ok, let's play the 'fifth' an octave lower. It now has partials at 75, 150, 225, and 300 Hz.
So the resultant set (added to our 100Hz note) is 75, 100, 150, 200, 225, 300, and 400Hz.
What's the difference? Well, 75 and 225 are new, and we've lost 600. But five of the original 6 frequency components (100, 150, 200, 300, and 400 Hz) are sill there. So there's still a lot of similarity between the new sound and the original one.
As I read about inversions, people seem to consider these to be the same chord, but in some inversion. But it's clearly not the same chord any more, since the intervals have changed.
Of course you're right in that it won't sound the same. But because
- notes are named in an "octave repeating" way
- two musical tones an octave apart will have some overtones in common, because a note's harmonics themselves are an octave apart
...there will be some similarity in the resulting frequency spectrum. The amount of similarity depends on the timbre of the actual notes you're playing.
So why even have the concept of inversions?
It's mainly an expression of similarity, rather than saying that they're actually "the same". They may be "the same chord" in terms of the name of the chord, but the fact that the name is the same shouldn't be taken to mean that they necessarily sound the same.
That's even true with chord names in other senses too - a 'C' chord in the bottom octave of the piano doesn't sound the same as a 'C' chord played in the middle of the piano keyboard, or on a guitar, or on an instrument tuned with a different temperament.