Can I get a tenure track Applied Math assistant professor position in a top US college if my PhD was in a similar ranked private U.S. college?
If you want to get your "teaching career" going, I wouldn't worry a lot about the ranking of the school. Teaching focused positions are going to look at your teaching experience, if you don't have a lot of this then you may need to take a visiting position to get some experience. 

A school ranked lower than the one where you got your PhD might view the name of your school as something they want associated with your department. Or they may assume you are more interested in research and not with teaching. These positions can still be incredibly competitive, but if you have the right background you may have a chance to be hired right out of your PhD.

If the school you attended was highly ranked, then schools ranked the same or higher are going to want most of their tenure track faculty to be focused on research, or at least to have a good research program to get undergraduates involved in. If you don't have a stellar publication record, you will absolutely need a postdoc. Even then, you will need a good postdoc with good research results. Even then, it will be very competitive.