[Vuejs]-In vue-electron, how to do you bind data between a parent and child components?

Electron is just a runtime for your web app.

Take a look at Vue.js’s event model.

https://v2.vuejs.org/v2/guide/events.html
https://v2.vuejs.org/v2/guide/components-custom-events.html

Also, check out component guide https://ru.vuejs.org/v2/guide/components.html

In your case

<el-button></el-button> is just a class added for the purpose of styling, not a parent component.

It is not a parent component it is a child component of <BookmarkMenu> component. And it may contain absolutely anything inside.

And this.$emit('add-bookmark'); method triggers an event listener (if any) of a component which would have <BookmarkMenu> as a child.

For example <BookmarkMenu @add-bookmark="someHandlerInAComponentContainingBookmarkMenu"/>