Friend 1's argument does indeed show that the circles have the same number of points, as each ray from the center uniquely determines a point on the circle, and each point on the circle uniquely determines a ray from the center.
Friend 2's argument (if made carefully) shows that the bigger circle has at least as many points as the smaller circle, but does not show that the smaller circle has strictly fewer points.
It all comes down to the notion of cardinality of infinite sets, where things can get pretty counterintuitive. If you want to give friend 2 an example to show why that argument does not show strict size difference, consider the set of integers $\Bbb Z.$ Obviously the same size as itself, yes? Ah, but now consider the function $f:\Bbb Z\to\Bbb Z$ given by $f(x)=2x$. This function maps $\Bbb Z$ into $\Bbb Z$, but misses infinitely many integers! That does not mean, however, that $\Bbb Z$ is strictly larger than itself. It is simply a peculiar quirk of (many) infinite sets that they can be put into one-to-one correspondence with subsets of themselves. Thus, showing that the smaller circle is in one-to-one correspondence with only a part of the larger circle isn't enough to show that the larger circle has more points--though such an argument would work for finite sets.
I think that it is not correct to say that "ancient hate the parallel postulate".
For sure, it is not so "self-evident" as others [but please, think at Common notion n°5 : "The whole is greater than the part"; until Cantor it was "absolutely" self-evident].
The possible explanation, as per Gerry's comment, is that it involves the infinite, and the infinite is not so easy to manage ...
According to Boris Rosenfeld, A History of Non-euclidean Geometry (original ed.1976), page 36, Euclid was "aware" of this :
Euclid tries to prove as many theorems as possible without using the fifth postulate. The first 28 propositions of Book I are so proved.
According to Rosenfeld [page 40] :
it seems that the first work devoted to this question was Archimedes' lost treatise On parallel lines that appeared a few decades after Euclid's Elements.
The title of this work in known only through the list of Archimedes' works by ibn al-Nadim (ca.990), and
it is possible that one of Ibn Qurra's preserved treatises on parallel lines represents an edited version of Archimedes' treatise.
[...] it is very likely that Archimedes used a definition of parallel lines different from Euclid's. [...] it is possible that Archimedes based his definition of parallel lines on distance.
Added
As finely remarked by mau, the original definition and postulate are [see Thomas Heath, The Thirteen Books of Euclid's Elements . Volume 1 : Introduction & Books I and II (1908 - Dover reprint) ] :
Def 23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.[page 154]
Postulate 5: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.[page 155]
Heath's edition comments at lenght definitions and postulates: the comment to P5 span from page 202 to page 220, with a lot of informations about the recorded attempt to prove it, from Proclus on.
Page 220 lists the most common alternatives to Euclid's version of the postulate; among them :
(I) Through a given point only one parallel can be drawn to a given straight line or, Two straight lines which intersect one another cannot both be parallel to one and the same straight line.
This is commonly known as "Playfair's Axiom" - from John Playfair (10 March 1748 – 20 July 1819) - , but it was of course not a new discovery. It is distinctly stated in Proclus' note to Eucl.I.31.
Best Answer
A circle is also a curve, a closed curve, indeed so why not?
In this link, red circle's parallel curves are shown and it also can be helpful: http://mathworld.wolfram.com/CircleParallelCurves.html. And I also suggest you to look at this: http://mathworld.wolfram.com/ParallelCurves.html