For the Celts, yes we do have a huge number of names being tossed about in that time. Celtae, Galatae, Keltoi (I believe in Greek it referred to their height), Gauls, etc. Remember that the Irish themselves have not even taken on a name for themselves at that time. They are still running around using tribal designations. The Belgae, in fact, used the term 'German' to describe themselves. In their case it appears to be meaning 'germane' or that they were the real Celts.
I think that we find similar situations with a number of groups. In the case of the Germanic peoples, I think we have very little evidence that there was any true pan-Germanic mentality. Today we can easily say that we have the East, West, and Nordic branches of Germanic, but I would imagine that there was very little in the way of recognition of kinship between each other. I say this even though my personal position is that, because their (The Germans) widespread expansion happened so much later that that of the proto-celts/Celts, the Germans are more closely related to each other than todays people of Celtic descent. The Germans too were more along tribal lines. We are prone to labeling and tend to look for the same trend among the ancients.
I find it important to reinforce that the restriction of the word Celt to apply specifically to those referred to as Gauls is a very modern outlook.
I normally date the trend at twenty years but we could push that to maybe thirty. Prior to that time, historical sources, probably with the realization that we needed some type of general name (labeling), used Celtic in a manner maybe akin to Germanic and broke down the specifics from there. There we would see Gaul/Gaulish, Brythonic, Goidelic, Galician, etc. The most recent source of which I can think that used the term Celt in the older manner would be Nora Chadwick's "The Celts".
The emergence of what I later found to be called the Atlantic school (Thanks to Cambria Red - I knew of the theories but not the name of the movement), which held that Celtic language/culture started in Iberia, was in my opinion, a major albeit unintentional contributor to the fray. Their work/positions and the trend towards restricting of Celt to Gaul, although both not necessarily coming from the same source, seem to have influenced each other. The two are more recent developments, going back about the same amount of time. It is possible that in order to work or cooperate with the position of Iberian origination of Celtic, a sort of default was required to explain how we had two possible ‘source’ locations of Celts. The result would be then to restrict Celt to the Gauls of central Europe.
The only thing appears to be 100% certain now is that this subject, being 20-30 years old, has while to go before it is settled.