Presumably all singularities are the same size: a point. What makes one black hole bigger than another?
Also, a singularity will only have the same mass (or less) than the collapsed star, so what accounts for the massive gravity (since gravity is created by mass).
The size of a black hole is defined by the radius of its event horizon, instead of the "size" of the singularity. Plus, a singularity is not a point, but a spacelike hypersurface.
As long as you are far away from the singularity, there is no difference that you can tell from a star. The gravity is the same. Gravity being massive is because the surface gravity at the horizon is infinite.
Hopefully, this helps a little bit.
The term singularity does not always refer to a physical object. It can also mean an end-point of logical reasoning that inhibits further progress in logical reasoning. For example, a result in our calculations that gives us an infinity or a zero where we think that shouldn't actually hold true.
I need to be very careful here to not give a solid conclusion about what a black-hole singularity is because nobody is really sure. The sure thing is that according to General Relativity in the ideal case of an uncharged, non-rotating black hole the co-ordinate distribution leads to an infinitely long line that would be the end of time (since time moves towards the singularity and carries everything at rest with it). It seems though that this type of black-hole is unrealistic and a rotating, charged hole is more likely. The singularity would then not be a point at all but a rotating ring making the model more complex since a ring would also have an interior.
What we know for sure is that mass does decide the size of the black hole and when mass doubles the radius doubles and volume increases eight-fold. You have a very good point in asking why the gravity is greater than the star that gave birth to the black hole. It is and it isn't. If you were to place an object where the original radius of the star was you would theoretically find that gravity had not changed. (This is probably not true as you also point out since the star might have lost mass in a supernova). The Event Horizon would be smaller than the original star's radius though and so the distance to the centre of mass would be less. Gravity has a stronger hold that way and it takes more energy to escape it. As you know not even light can escape.
However gravity would theoretically continue to increase in the interior becoming maximum at a singularity if there is one.
