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As mentioned in a comment, you seem to wish to classify sets of points, not graphs as such.

If you wish to classify them bijectively with word, you need the cardinalities of the sets to match; that is, there must be the same amount of possible word and the same about of possible sets of points.

The cardinality of the set of words is countably infinite, or if you set a bound on their length, some finite number.

If you wish to make the cardinality of sets of points countable, then you certainly need to discretize the plane (for example, only use coordinates with integer values).

If you wish to make the cardinality of sets of points finite, then you need to have an upper bound for how many points to place and you need a finite set of coordinates they can be placed into. 
Then, if you really need a bijection, you need to have an equal number of possible words as of possible sets of points. But maybe it does not damage if there are more words, so that not every word corresponds to a set of points?

In any case, once your two sets have the same cardinality, there exists a bijection, and likely quite a lot of bijections, between them.

The words have a natural order in terms of length and alphabet, so if you can create a natural order for the point sets, too, then you are done. Maybe order them by the number of points first, and then for example left to right, down to up (in case of a bounded part of the plane) or starting from closes to the origin and then in a clockwise fashion. Really, any ordering for the individual points, plus the idea of alphabetic ordering, does the trick here.