I’m kind of surprised by this question, because it makes it all seem a lot more mystical than it really all is.
Every group has what’s known as “signature sound.” A signature sound is something they do that’s particular to them. So as soon as you hear a song of theirs, you go, “Ah! That’s so and so.”
So what makes a signature sound?
1) Mood. Some bands have a particular mood to their music-- like upbeat, downbeat, sexy, romantic, childlike, whimsical, melancholy, laid back, peaceful, agitated, etc…
Example: Sugar Ray. It’s so easy to identify the music because there’s a very distinctive, “mellow,” laid back feel to it that so few other bands have.
2) Style of playing. Some bands may have played the same instruments, but not all of them played them the same way. In fact, some played their instruments in such a unique way that it’s the style of playing that automatically identifies them. For example: George Harrison. His guitar in his earlier work had this distinctive “wah wah” where as soon as you hear it you think, “Oh, that’s definitely Harrison.”
The Doors are another example. The keyboard playing was just distinctively “Doors-ish.”
More modern day example: Coldplay. What other band plays piano like that? Only Coldplay.
**3) Chord and chord progressions. **I’m not even going to pretend that I’m a major scholar on this, but my gut instinct tells me that there was a particular way that groups used chords and progressions that’s a large part of the “signature sound.” One article I came across said that what made The Beatles sound so distinctive at the time they debuted was that their chord progressions were a lot more sophisticated than what was common at the time. Another argued that they threw in a lot of “odd keys” in the middle of the new ones. Argument aside, there seems to be a consensus that they were doing something with chords and progressions that disinguished them musically and made them sound “Beatles-esque”.
4) Choice of instruments. I think it’s oversimplifying things a bit to say that all groups use the same instruments. Not every group uses the same standard instruments. In fact, the usage of a particular instrument that no one else is really using plays a large part in creating a “signature” sound that identifies a group. You know it’s Stevie Wonder when you hear the harmonica. You know it’s the Doors when you hear the keyboard. You know it’s one group or another if you hear a flute or a theremin or banjo or a cowbell because only that group tends to use it. Groups use non-standard instruments more than you think.
5) Fluorishes. There are some “personal touches” that groups use that identifies them… Perfect example: Fatboy Slim. DJs were a dime a dozen back in the '90s. But he did this thing with almost every song he made where you instantly knew it was him-- at one point, he would play a thumping bass very slow, then accelerate it rapidly. You heard that and you were like, “Oh, it’s Fatboy Slim.”
Other fluorish examples: using a vocoder, ending or opening a song a particular way, etc…
There’s probably a lot more to the “signature sound” to this (types of rhythm, tempo, etc), but these are the ones that instantly come to mind.