The appearance of the toolbar button is different to the built in ones. Presumably you mean design features out of your control, so not your choice of icon?
You need HTML to put in the popup. Sadly you’ve left out what this HTML should do and look like when rendered.
You offered us an excerpt of your manifest, which I take as problem, since you think there’s something wrong with it and you expect a change to it may solve something, either a third issue or one of the two above.
Plus you’ll always have an arrow pointing at your button to make it very visually apparent that this is not a native menu list.
Much more importantly, it will work very differently from an accessibility standpoint, since you’re an HTML context, so you get HTML keyboard navigation and accessibility features etc. and not the native menu features.