This is a follow-up to two chapter discussion post which you can find
here and
here respectively. Since what I want to say has nothing at all to do with that chapter, but very much with what has been written in this thread, I'm going to put it here.
The ring-giving as a gesture of a ruler towards his subjects is something Middle-earth and the historical world definitely have in common. In the instances where a ring is passed, even in a mood of gratitude, such as Barahir's ring, it is still a sign of fief, demanding mutual keeping of terms such as protection and service in arms. Sauron took this one step further with his rings, especially the Nine.
In this light, Sauron showing up in Eregion and passing out rings, even in disguise, is bold, if not insolent. The underlying intention of ensnaring the Elves must not have seemed so underlying at all. The Three, in contrast, might appear more as an effort of the Gwaith-i-Mirdain to claim a leading rôle among the Elves, or rather, to strengthen those Elves which already had a standing of their own in their position. Still, I'd say the Three are more like powerful artifacts of which one is just a keeper; the power of the rings themselves and the defiance of Sauron for which they stand are more important.
Sauron creating a ring for himself can certainly have many implications, just read above for some musings on the matter.
What is striking is that by forging and giving a ring to himself, he states that there is noone else to give him a ring, i.e. noone who could be his lord, noone who could be above him. In that regard, merely forging the ring to keep for yourself is an act of proud defiance of Eru; taking it as a ruling ring even more so.