You would also see it named N,N'-dimethylmethylethane-1,1-diamine. Both this and the 1,1-bis(methylamino)ethane would be correct.
In this case, 1,1-dimethylaminoethane should also lead you to the same structure, but only because the other interpretation gives 1,1-dimethylethane, which is an incorrect name (it would be 2-methylpropane). It is still a little ambiguous on first glance. Typically when you are naming functionalised or branched substituents, you would separate their naming into brackets so as not to confuse with the naming of the parent chain. It is also for simplicity. When you have larger molecules that have a lot of functionality and branch points, having the various bits in brackets makes it easier to construct a mental image of the structure. For example:
The above is named 2-chloro-N-(4-chloro-6-methoxypyrimidin-2-yl)pentanamide. Without the brackets there, the numbering on the pyrimidine ring becomes ambiguous, since there is no clear way to tell if the numbers refer to the ring or to the parent chain.