Our modern conception of a religion as a distinct group that you belong to exclusively is something that has developed rather recently in the history of religious belief and actually coincides with the rise of Christianity and Islam in the West. In fact, in countries like China, such a distinction still doesn't readily exist. During "Pagan times", what we think of as "religions" were typically divided along cultural lines; certain cultural groups had certain deities, rituals, and beliefs about the world around them. However, there was not an idea that because you were Greek you could not worship Anubis or Odin because they were not part of "your religion", rather deities were commonly syncretized or traded when two cultures came into contact.

The closest thing to "religions" as we know them today were mystery cults, special religious groups with voluntary membership that provided spiritual fulfillment for those whose needs were not being met by the overarching state cults. These groups often purported to have secret knowledge that was only shared with initiates and promised direct experience with divinities in addition to benefits in the afterlife. These groups were rarely exclusive, however, and many people, such as the Emperor Julian the Apostate (the last Pagan emperor of Rome) were initiated into multiple mystery cults.

This changed when Christianity became popular during late antiquity. Christianity, unlike the mystery cults, specifically marked itself as in opposition to all other religions; they believed that there was only one God, only one way to worship Him, and that all other gods were lies or demons. Since they were an exclusive group, they were identified as one. Towards the end of the Pagan era, the Emperor Julian did try to reorganize the state religion of Rome into an organized religion along the lines of the Christian church and referred to this religious group as "Hellenismos" ("those who worship in the style of the Hellenes" in Greek).

This is something of a simplification and does not get into the continuing development of religion as an identity that occured once Islam rose in the Middle East and Christianity found itself faced with it's first organized opposition in centuries. If you are interested in the changes in the religious structure of the West during the fall of Paganism, I suggest checking out "God Against the Gods" by Jonathan Kirsch.

Groups/tribes/peoples all over the world had names for themselves, but often in small scale societies the names for themselves just translate as "the people", "the real people," "the free people," "the original people", "the brave people", etc. Take the Germans, for example. Our word is untraceable past the name of the Germani, a specific tribe. But in German, they're called Deutsch, which is related to a Proto Indo European root meaning people. The Germans in French are les allemands, which is ultimately likely related to "all men". French, for example, comes from the Franks, who were the free [not servile] ones. Swede originally either meant free or kinsmen. The Navajo word for Navajo is Diné, meaning the People. The Cree either call themselves "Nēhilawē, those who speak our language, or Eeyou, the people. The Guaraní called themselves Abá, meaning men/people. This pattern is repeated throughout the world, though obviously it doesn't apply to every single group of people. Other names might be like "those who live by the rapids/in the forest/on the plains/to the north" or "those who eat [the thing their neighbors don't eat]/do [what our neighbors don't do]", etc.

Just as "who we are" is distinguished from "who they are", "what we do" is often distinguished from "what they do". The term religion, and particularly in its current meaning, is surprisingly recent, only coming out of the nearly simultaneous events of the European discovery of the Americans and the schism between Catholicism and Protestantism (see J. Z. Smith's "Religion, Religions, Religious"). But even though the term is new, what we'd call religion existed much longer. But they were probably differentiated similarly to people, our religion was "the way of gods" or "our path" or "the true law" or "the original custom" or "knowledge of the spirits" or something like that. One of the problems is that for most groups, writing and contact with one of the great proselytizing religions (Buddhism, Christianity, Islam) is almost simultaneous, so a lot of the records we get are through non native eyes. I can't remember how the Sumerians and Egyptians talked about their religions, but the Dao Te Jing, a fundamental text in Chinese religion (especially Daoism, of course), says famously, "The way that is spoken is not the way. The name that can be named is not the constant name." But, here "way" is literally "dao", so another way to translate this line is "The Tao that can be told is not the eternal Tao." The word "veda", used in Hindusim, means "knowledge". "Torah" means "Law". Dat, the Modern Hebrew for religion, originally meant "law" or "custom". And so on.

Since I know some groups, even some small scale societies, have more specific names for themselves than "the people" or "the free ones", I can feel confident that some groups had more specific names for their religions than "our customs" or "the true way", but I can't think of any off the top of my head.

It's difficult to ascertain, but seems unlikely, as the forms of pagan worship that we have records did not keep a written word until after the introduction of Abrahamic monotheism, whose writers would have recorded as simply an equivalent for pagan, heathen or infidel.

Records of paganism would more describe the peoples who practised it, so a Scandinavian (Norse) who practised Pre-Christian beliefs was a norse pagan. It seems likely that the pagan followers would have simply refered to themselves as the peoples or region they came from, and they would practice the local beliefs and traditions. With the introduction of the new Abrahamic faiths, they might refer to themselves as something "following the old ways".

Names such as Asatru (Norse), Romuva (Baltic), Suomenusko (Finnic) are the names given to the modern Neo-paganism movements, which usually claim some form of continuum from the pre-modern beliefs, rather than the historical forms. These names are created and used by neo-pagan followers, often which may translate to something like "The Way".

Davies, Owen (2011). Paganism: A Very Short Introduction. New York: Oxford University Press. ISBN 9780191620010.

Robert, P. & Scott, N., (1995) "A History of Pagan Europe". New York, Barnes & Noble Books, ISBN 0-7607-1210-7.

Blain, Jenny; Ezzy, Douglas; Harvey, Graham (2004). Researching Paganisms. Oxford and Lanham: AltaMira Press. ISBN 978-0-7591-0522-5

Lewis, James R. (2004). The Oxford Handbook of New Religious Movements. London and New York: Oxford University Press. ISBN 0-19-514986-6