Pumpkin spice products generally don’t actually contain pumpkin - they contain the spices used in pumpkin pie, usually at least cinnamon and nutmeg, but also could include things like cloves, star anise, cardamom, allspice, and ginger. These warmish spices pair really well with hot coffee beverages and cream, hence the popularity of pumpkin spice lattés. But it’s not just lattés - you can also buy pumpkin spice flavored coffee creamer to use in a regular cup of coffee.
In the States (or at least the Northeast where I’m from), pumpkin spice is sort of emblematic of the start of Fall. As the weather turns cooler we get into the spirit with things like big comfy sweaters, decorative gourds, chrysanthemums, and scented candles (apple pie and pumpkin spice scents are popular!), but spiced food is also a huge part of it: for example, gingersnaps, mulled apple cider (which uses a lot of the same spices as pumpkin spice), and of course pumpkin spice lattés.
A lot of these spices are used throughout the rest of the year because we’ve got Thanksgiving in late November (pumpkin pie is a staple for this holiday), and for Christmas there tend to be spiced cookies and cakes, plus spiced tea and mulled wine. And there are also spice-scented air fresheners and potpourri, so whether you’re at home or out shopping there’s a nice warm cozy scent.