You're talking about porting your mortgage, which may be possible if your mortgage was portable to start with, or if your bank subsequently allows it.
Note that although porting a mortgage involves keeping most of the original terms and conditions, the process is still much like applying for a new mortgage, including any lending requirements.
Here's an article on the subject.
EDIT: In response to OP's comment below:
What will happen to the first property if I don't sell it?
Because porting a mortgage is treated as if you were closing one mortgage and opening a new one, this means that you would need to pay off the first mortgage.
Typically this would be done by selling the first property at the same time that you buy the second one. However, if you're not doing this, you'll need to raise funds another way, which could include opening a new mortgage on the first property (of course, if you're doing that, then there would have to be a good reason for porting the original mortgage; otherwise you might as well leave it where it is, and open a new mortgage on the second property instead).
Does the article apply the to USA too?
That article (and indeed this answer) are based on the situation in the UK. However, they appear to exist in the US too, though are rarer than in the UK.