One interpretation often given is that if a plane surface contains two points, then it contains the line connecting the two points. If that were the meaning, then it would be just as well to make that the explicit definition or to make it a postulate. But that does not seem to be Euclid's intent. His proposition XI.7 has a detailed proof that the line joining two points on two parallel lines lies in the plane of the two parallel lines. No proof at all would be necessary if that line were by definition or by postulate contained in a plane that contained its ends.
Note that a plane surface may be infinite, but needn't be infinite. It can be a square, a circle, or any other plane figure (Def.I.19).
There are no postulates in the Elements for the existance of plane surfaces, either finite or infinite. Post.3 says circles can be drawn, but a ambient plane is implicitly required there. Rectilinear figures are assumed to exist once the bounding lines have been constructed, but again, a plane is presumed to exist first. Throughout Books I through IV and Book VI, the books on plane geometry, there is the implicit assumption of one plane in which all the points, lines, and circles lie. In the books on solid geometry, Books XI through XIII, there is sometimes mentioned a "plane of reference," and proposition XI.2 claims that two intersecting lines determine a plane as does any triangle (but its proof fails completely).
Next definition: I.Def.8