The interval between C and D♯ is an augmented second. The interval between D♯ ad F♯ is a minor third.
Both augmented 2nd and minor third intervals sound the same.
If we called the first notes C and E♭ (enharmonically the same notes, then that would be called a minor third.
If we called the second set of notes D♯ and Ex (E♯♯), that would be an augmented second.
There are several ways to name intervals, depending on what names each note is given. Just listening in isolation, it's virtually impossible to say what the notes are called, thus virtually impossible to give the interval between them a name.
It's hardly any interval sounding the same, but several.
In your case, one interval is an augmented second, the other a minor third - both of which sound the same. Main consideration for intervals is deciding what letter name to call each note. For example, had you called the 1st C, the next E♭ and the last G♭, there would be two minor thirds - C>E♭ and E♭>G♭.
Playing them all, as a chord, would then make C diminished triad.
Edit: now you've pretty well changed the question, my answer, which addressed the original, is pretty well redundant.
There is a straightforward scientific reason, which the other answer provides. But the whole point of having these things musos call 'intervals' is that yes, they will sound the same - they're supposed to.
Let's take P5 - a perfect fifth interval. It could be C>G, D>A, B♭>F, G♯>D♯, and many others. It doesn't matter whether they're low or high notes - as long as the second is 7 semitones above the first. And all the other intervals - and their equivalents - sound the same because that's what they do - physically down to ratios of frequeny - which stays the same for any given interval (and equivalent interval) regardless of which octave the two notes are from (high or low, but they must both be in that same octave). The phenomenon was discovered, and is a useful piece of knowledge
However, you can still take note of my previous answer, as that tries to explain something which you appear to be somewhat vague about - the naming of intervals themselves.