Although "musical significance" can be a turn of phrase used without really realizing it, musical significance is a thing, and it can manifest itself in several ways.
Sheer innovation is musically significant, especially if the innovation inspires other composers to follow suit. Musical significance based on influence on later composers is often only recognized long after the initial innovation.
There's a big trend in music theory today called schema theory. Schema are "stock musical phrases" (read more here) that occur often enough to have been given names for easy identification. I think this is actually what the OP of your linked question was asking, just in very different words: is this particular feature of a progression a common schema?
Motives are often musically significant because they recur time and time again. Mahler was great about hiding little motives in an accompaniment and then later developing them into full-fledged themes. Like the "innovation" heading above, such musical significance is not realized until later in the piece.
The list could go on and on; this is obviously a very incomplete list. But hopefully this gives some idea of what can be meant by "musical significance," since it's a tough thing to define without saying "it's something that's musically significant."