Among the flavours of onions, spring onions, shallots, leeks, and chives there is one that they share. Is there a name for it?
Conversationally, "oniony". Everyone will understand that, and tend to say it naturally.
In a more serious cooking context, you could also get away with "allium flavor", though likely not in everyday conversation. There are a lot of alliums, including onions, garlic, scallions, shallots, leeks, and chives, so if you refer to the whole group, you're pretty obviously referring to the overarching common flavor. For example, On Food and Cooking has subsections called "The Flavors and Sting of Raw Alliums" and "The Flavors of Cooked Alliums".
"Alliaceous" is the adjective to describe the group of plants, in a botanical sense. I use it to describe the flavour as well, but it's worth noting that a lot of people would not know what that means straight up. That might be similar for "allium" in general though.
This is sometimes called the pungent flavour.
The term pungent vegetables is usually used to specify garlic, onion and leeks in diets that omit them, such as Su Vegetarianism. This is also a nicer sounding name than fetid vegetables, which is apparently behind the etymology of asafoetida, a supposedly similar flavour.
In East Asian Vegetarian diets, they are named as "The Five Pungent Spices" and are being forbidden from consumption. Note that, other than Allium vegetables, the word also include Asafoetida for Buddhist, Coriander and Brassica Rapa for Taoist.