What do all the distributions that have the same Gini index have in common?
This is a Lorenz curve.



The Gini coefficient is the area between 'equal wealth' and your chosen line, divided by all the area under the equal wealth curve. There are other illustrations online, for sure, which would explain this.

The Wikipedia article is telling you that the actual distribution of income is not fully captured by the Gini coefficient. The same Gini coefficient could have different distributions of income. For example, it could move equally in the lower region at first and then flatten between 60 and 80 per cent of people. Or it could be that poorer people get nothing and the middle and upper classes are hoovering up the income. 

Both examples, if you set up the toy numbers properly, imply very different distributions for poor people, but produce the same Gini coefficient.