My question is specific about ice-creams but more details are welcome.
It's a flavor. It's on the subtle side, particularly in the quantities it's often used in, and maybe if you've eaten a ton of vanilla ice cream you don't notice it anymore. (Or maybe you just haven't had very good vanilla ice cream.)
The flavor is either from the vanilla bean if it's fancy vanilla ice cream, or more likely from artificially produced vanillin (the main component of vanilla's flavor). The aroma is pretty distinctive; the best way to demonstrate to yourself what it is is to find a bottle of vanilla extract. Artificial vanilla flavor is much, much cheaper, and in a lot of contexts is hard to tell apart from real vanilla, so it's very common.
Same goes for other desserts; vanilla or vanilla extract is a pretty common ingredient. Sometimes it's the whole flavor, like in vanilla ice cream or pudding, and sometimes it's one of many, like in a lot of cookies. It pairs well with a ton of other flavors, including fruits and spices, and it blends well into the background, rather than covering up other flavors, hence its wide use in desserts. That's also why vanilla ice cream is so handy: you can eat it with just about any other dessert.
This is presumably why it's taken on connotations of "plain". It goes with so many things, and even if it doesn't really positively pair with something, it probably won't taste bad with it, and might just get covered up. So in practice, that means you can treat it as the "plain" flavor of ice cream, especially since so many not-plain flavors of ice cream start with a vanilla ice cream base. It provides a good background flavor that works well as a plain but not boring starting point.