If I wanted to create a game like Fallout Shelter that is viewed from the side in 2D, but is actually 3D.
Should I still use the 2D view of Unity or use a 3D view and move the camera? And somehow build everything maybe from a top down perspective with characters lying down?
Just try and find what is better for you. This is your game design.
I suggest 3D if the characters will be in 3D. If they are sprites - in 3D leads to other issues with lights and shadows, but not unsolvable.
However consider a better reference instead of Fallout Shelter - how about Trine? 
I agree that more references would be helpful. If you decide to move forward with a 3D camera you’ll benefit from the inherent depth built-in to a perspective camera. Other issues can arise if you decide to overlay GUI elements over models in 3D as opposed to using screen space. It’s important to prototype the camera systems for any GUI and game elements before you get to deep into it. It’s not all that easy to just switch a camera from perspective to orthographic and keep the same look.
I am currently working on a 65" touchtable game that is rendered in a perspective camera with GUI elements on 3D models, the perspective shift at the edges of the screen aren’t desirable but it the depth outweighs the issue. I have considered rendering my player objects in a separate camera but decided against it so far.
Anyway, prototype first to make sure it looks the way you want it to. Always happy to help if you have more specific questions. Good luck!
Maybe run through some relevant tutorials to get an idea of how it all differs. Have a read of this.
Thing to remember is that you’re always in 3D space even when working in 2D. It’s just that 2D streamlines some things you don’t necessarily need to worry about with 2D movement. You can mix 3D and 2D assets if you want.
Not actually played fallout shelter but from what I can tell it uses a mix of 3D models and 2D sprites with a 3D perspective camera. The rooms look like 3D geometry and characters look like 2D animated sprites. Other stuff like furniture, props, surrounding ground, etc look mixed as well.