That's not true: you make the SID sound weird and exotic when it's really not (although it is incredibly cool).
Seen from the perspective of microcomputers, which mostly either used simple square waves, or FM synthesis to generate sounds, the SID is unusual. But seen from the perspective of sound synthesis the SID is an analogue subtractive synthesizer[0] which, at the time of its design, were commonplace. As I say, that was pretty unique amongst microcomputers, and is why the SID was and is revered by fans of the 8-bit era: it has a really fantastic, crunchy analogue sound, that other computers of the era just don't and, indeed, can't reproduce.
I'd go as far as to say that Commodore missed a massive opportunity with the Amiga in terms of use for music: if it had the PCM based Paula chip, plus a SID chip (or even two - a man can dream), it would have been absolutely incredible.
In recent years analogue synthesizers have experienced something of a resurgence although, in truth, they never completely went away. As such ICs with analogue circuitry are actually relatively commonplace. Now that doesn't mean you can just remanufacture the SID chip really easily, but it is absolutely possible to build ICs with analogue circuitry today: as a trivial example, analogue to digital converters are commonplace.
[0] I think strictly speaking it's also a paraphonic synth, since all the voices share a single filter, but I can't remember off the top of my head whether they also all share a single amp (again, a hallmark of a paraphonic, rather than a true polyphonic, synthesizer).