What is the style called where we draw characters without lifting the brush?
I would just call it 一笔画 (one stroke).
Because there are some famous arts are called in this way. Like:

一笔虎: write the character 虎 in one stroke.
一笔龙: write the character 龙 in one stroke.

And the Eulerian path is also called 一笔画问题 (the problem of how to draw a graph with one stroke)