No, it really shouldn't.

The accessibility guidelines we have, e.g. WCAG, usually say that tab is used to tab through separate interface elements. When you're still inside an element - for example if you have triggered a dropdown list in a text input field - you use the arrow keys to navigate within and enter/space to select elements.

Say tab was used for all (intra and inter) navigation tasks. In that case you risk suddenly having a thousand item list that keyboard users will then have to tab through to get to the next interface element. That is not good UI design.