No.... the identity seeking for 4 is over-emphasized. 4s envy the inherent significance others have that they feel they are missing. Emotions signify stuff - they add weight. So feeling intensely makes everything, well, feel more significant. This is substituted for when there is lack of real meaning, and is often done with fantasy when real life feels dull & the 4 feels insignificant. It's not an identity, but significance being sought. All image types seek to create an identity which helps them cope with shame. 2s have one of being loving or desirable, having a lot to give. 3s have one of being impressive, successful, capable. 4s have one of being unusual, refined, elite. Those are generalizations, but the identity sought connects to the source of shame for each, which you note for 2s & 3s, but miss the shame motivating the 4. The 4 feels missing of something others have, so they try to turn the "different" aspect into something which fills what they are missing - that individual significance.
This is where the 4 drama lies - not in a show, but in amplifying feelings to make things seem more significant, including their experience of themselves (I'm not boring; I feel intensely).
IMO, lots of 7s seem very emotional & intensely feeling. I think they amplify it for different reasons - they seem to fear an inner void, facing silence, etc. If it's too quiet, then maybe the deeper anxieties surface. I can relate to that a bit as a 4, as far as feeling a void, but it's more the idea of having no notable personality that shames me. I'll think there's nothing there that would mean anything to anyone, and the emotion & fantasy is just kind of a comfort for the real life I may never have.