Does "having" something imply the possession of it?

Solution 1:

There's a difference between the present continuous ("to be" + present participle in "-ing"), and other uses of the present participle.

So, in your example:

...-some (having a particular quality)

there is no "to be", so this is a use of the participle as an adjective, not a present continuous. It is equivalent to saying:

...-some (that have a particular quality)

You are correct that using the present continuous here

*that are having a particular quality

would be inappropriate.

Similarly, in your "friend" example, you are right that saying

I'm having a friend

that could have a number of meanings(!), but not that the person in question simply is your friend - but, on the other hand, saying

People having friends in America knew more about the issues surrounding the president.

is fine, and equivalent to

People that had friends in America knew more about the issues surrounding the president.

Well, you could certainly use "having" to mean possession. Consider:

I got so used to having a car that I was unprepared for how arduous it was to walk to work every day.

Having 500 friends on FaceBook isn't as good as having two actual friends in real life.

What does having high cholesterol suggest about a person's dietary habits?