Rusty beat me to post an answer, but I guess one more answer won't harm.
No, it never means "to possess".
"Había algo ahí" and "no había la más pequeña brisa" are impersonal forms to say something existed. Neither the "something" nor the "breeze" were possessed by any subject.
The impersonal conjugation of "haber", which has in present a distinctive conjugation --"hay", means "there is"/"there are"; in this case, "there was (not)"
- En este foro hay buenas personas.
In this forum there are good people.
- Hay una araña en la pared.
There is a spider on the wall.
- Hay un niño gritando afuera.
There is a child screaming outside.
- ¿Hay buenos libros en tu biblioteca?
Are there good books in your library?
- No había la menor duda de que estábamos perdidos.
There was no doubt that we were lost.
- Había una luna brillante en el cielo cuando desperté.
There was a bright moon in the sky when I woke up.
- ¿Hubo muchos invitados en tu boda?
Were there many people invited to your wedding?
- ¿Crees que habrá clases mañana? Está nevando muy fuerte.
Do you think there will be class tomorrow? It's snowing very hard.
- Habría más escuelas si los políticos gastaran menos dinero en guerras.
There would be more schools if politicians spent less money in wars.