EMF of an EMF source (a battery for example) is defined as the work done by the non-conservative force(s) on charged particles as it passes through the terminals of the source divided by the charge of the particle. That is $\epsilon=\frac{dW}{dq}.$ This definition is very similar to that of potential difference. But the major difference, among others, is that the force, in this case, is of non-conservative nature while the potential function is defined only for conservative forces only.
The potential difference between two points is independent of the charge of the particle ie the same potential difference will be calculated between two points whether we calculate it using a $10C$ or $-1.9C$ charge and also independent of the path the charge take. But this is not the case for EMF. The EMF depends on the path and the charge on the particle as it is defined for a non-conservative force. Hence EMF of a source calculated using $10C$ particle along a path will be different if the particle took another path and it will be different still if we move $-1.9C$ charged particle along the same path. So what do we mean when we say that a battery has EMF $\epsilon$? How can we be sure that the work done by the battery-by the non-conservative forces due to battery-on a charged particle $Q$, as it moves through the source, will be $Q\epsilon$? Won't it be different if it took other paths?