Is the event horizon the defining characteristic of a black hole?
Yes, it's the only widely accepted definition.
This definition has the disadvantage that it's nonlocal. That is, it defines a black hole based on whether or not light can escape to infinity, which means that you can't judge it based on the properties of the spacetime near the black hole. So for example, if you're doing a numerical simulation of gravitational collapse to form a black hole, there is nothing you can look at easily in your simulation to define where the event horizon is or whether it has formed yet.
An advantage of the standard definition is that for an object that is a black hole by that definition, we can prove no-hair theorems. Observationally, it also connects directly to criteria we can use to tell whether a given object is a black hole.
There have been attempts to come up with alternative definitions: http://arxiv.org/abs/gr-qc/0508107
Or can I say something is a black hole if it has an equal gravitational strength?
No, this won't work because there is nothing special about the gravitational strength of a black hole (assuming you define that as the gravitational field g at some fixed distance). For example, if a star undergoes gravitational collapse to a black hole, the strength of its gravitational field, at a fixed distance, is the same before and after collapse. The equivalence principle also tells us that the gravitational field g is not a very useful concept to work with in general relativity, because it can have any value you like (including zero) depending on your frame of reference.