Amber has no scent
There are three things called "amber" associated with perfumery.
The first is ambergris. It's produced in whale stomachs, possibly to protect them from hard sharp objects (squid beaks, for instance), hence the nickname "whale vomit", although "whale pearl" might be just as accurate. Fresh ambergris is a byproduct of the whaling industry, but considered unsuitable for use in perfume; the good stuff spends years in the ocean before finally washing up on shore. I haven't smelled the real thing, but the synthetics have a distinctive fresh-woody character with some resemblance to clary sage; the real thing is said to be smoother, more complex, and somewhat animalic.
The second is mineral amber. This is fossilized tree sap. It's translucent and kind of an orange brown color, like honey; it is, I suspect, what the color "amber" refers to. Actually, this isn't used in perfumery at all; mineral amber has no smell unless it's heated (by burning, or working with power tools) and the smell is supposedly an unpleasant resinous or "burning plastic" smell. According to some sources on the internet, it's possible to extract an oil from the resin, but the smell is similar to when it burns.
The third is perfume amber, which is often confused with mineral amber. Perfume amber isn't any kind of fossilized sap; it's a solid perfume, typically composed of labdanum, benzoin, and beeswax, with any number of other ingredients as the maker desires (patchouli, frankincense, and vanilla being probably the most common). The scent is heavy, resinous, and sweet, but not particularly like ambergris. It's also possible to extract an amber oil from the solid, or to construct an amber oil by using extracts of the resin ingredients.
So when a scent says it has "amber" notes in it, which of these three do they mean? Unfortunately, there's no way to tell. Ambergris has a kind of legendary status, being known as a precious material that makes any perfume better but is too rare and expensive to use in modern commercial formulation.. a kind of unattainable exotic luxury from the past. The word "amber" has an evocative, mysterious quality that makes it sound desirable even to people who don't know what it is. No perfume marketer would hesitate an instant to claim that something has "amber" or "ambergris" notes; their real stock in trade is genuine 100% Grade A bull****, and they'll say anything, anything, to sell a bottle of juice. But since ambergris has its own word, I generally assume that when a perfume says it has amber in it, they mean the solid perfume kind.. and I think most perfumes with "amber" in the name have the characteristic smell of perfume amber.