I'd like to preface this by seconding Sees' observation that Paganisms, contemporary or otherwise, are very theologically diverse.
Is every individual deity considered to be a unique being, or is it that some peoples and religions recognize the same deity under different names? I have heard of Mars and Ares as being basically the same god, for example. Would the different Sun gods of different religions actually be considered different interpretations of the same Sun god?
The only blanket statement I'd feel comfortable making with respect to how Pagans approach the gods is that they do so in a way that reflects their experiences of them. I think Sees does a great job of presenting certain aspects of this, but there's a specific take on this that I think is worth discussing.
In contemporary Pagan circles, you will sometimes see discussions about two flavors of polytheism: "soft" and "hard." Generally speaking, "soft" polytheists would not regard various cultural deities to be unique or distinct beings, but different interpretations of some underlying archetype or unified force. "Hard" polytheists on the other hand - whom are more representative of Paganisms of antiquity - would
definitely regard various cultural deities to be distinct.
Some of our community leaders have speculated that as Neopaganism matures, it's polytheistic theology will more closely resemble that of antiquity. This is based on the hypothesis that soft polytheism is a byproduct of Neopaganism arising out of a predominantly monotheistic culture. People converting away from monotheistic religions can't quite fully remove themselves from that one-god idea, so they meet polytheism halfway and are "soft." I think there is at least
some truth to this observation, but I hesitate to put too much weight behind it. There's certainly no denying that contemporary Paganism is inevitably influenced by the dominating monotheisms of our culture, but I'm not sure I favor the slightly condescending attitude that "mature" Neopaganisms are hard or "true" polytheist.
Honestly, given that Paganisms are more about practice, the theology behind worship is largely irrelevant. Pagans, for the most part, approach various gods as distinct individuals regardless of whether or not they are "soft" or "hard" polytheists. It's akin to approaching specific humans as
individuals instead of as representatives of the collective essence of the human species.
If there is no such thing as pure fantasy, does that mean that every idea and character has a life and existence of its own? What if a person "invents" a god? Is that god then something that can be worshiped? Or would it instead be considered that the person "discovered" the god instead of inventing them?
Nobody agrees in the answers to these questions in the community, so I'm not sure I even want to
attempt to represent all the perspectives out there. I like to keep it simple: gods are that which someone regards as worthy of worship, for whatever reasons they deem sufficient. That which can be experienced in
any way by us is part of our reality, and may be deemed as a subject of worship. I personally do not concern myself with fussing over what aspects of reality are "more real" than the others. It's all real. Whether or not something was "invented" or "discovered" is irrelevant: it's all real, and it can all be worshiped if you choose.
In a fashion, this touches on one of the most important characteristics of Paganisms: you pick your gods, and you worship what you want. Whether or not someone else believes in or approves of your gods and your worship of them is the sort of intolerant nonsense that was born out of exclusivist religious positions, not polytheistic theology.
How would anthropomorphic concepts like Old Man Winter, Father Time and Santa Claus be regarded? Would they be considered gods, spirits or something else?
I'm not entirely sure what to make of this "anthropomorphic concepts" business. That categorization seems somewhat arbitrary and I'm not sure I understand what you're putting into it - and what it is contrasted to. Human understanding of the gods is frequently depicted in anthropomorphic terms in all theistic religions. It's poetic metaphor and artistic license; the "concept" is understood to refer to an actual
thing out there, as filtered through human understanding. I'm also confused as to why Santa Claus is on the list, because he really doesn't seem to be of the same sort as the other two.