Generally when folks equate two deities they equate deities of the same gender. Hence why people don't tend to say that Hera is the same as Indra or that Isis is the same as Apollon. However, by saying that Amateratsu is the same as Ra you 'break' this 'rule' and therefore have to open yourself up to other interpretations. If we are equating gods only on what natural phenomena they are associated with, why don't we equate say Zeus and Pele? Maybe Brigid and Hephaestus? Even equating based on role and function can get kind of messy, Brigid and Hephaestus are both smithing Gods, can we equate them on these grounds? I think most people would say no.
Then what does it mean for these Gods who rule over domains that there is only one of? How do we potentially reconcile this dissonance? Well, I don't know how much this will help you, dear reader, but this is the ways in which I can reconcile it.
In the case of ruling over things which there are only one of, such as the Sun, or ruling over phenomena, I think it is best to look at those things as symbols of the God's power. For Ra the Sun represents his immense and mighty power, for Helios the Sun represents his watchful eye (and those two examples are way oversimplified, I know.) I have seen folks use the symbolism explanation for Zeus's association with lightning and rain taking those as symbols for his might and fertility (respectively). I just don't see why they can't do that for Gods so obviously different as Ra and Helios and Amateratsu.
In the same vein as this, they could very well 'share' the role. The example I saw used was Hades and Anubis, does that mean that there are different afterlives assuming of course, that they aren't the same. I'd have to answer with a resounding 'yes'. We don't give the gods a lot of credit sometimes, is it so hard to believe that beings with the power and ability to act with agency on our lives wouldn't be capable of 'creating' multiple afterlife scenarios? Does that mean I think that there is an afterlife for each and every Death God and Goddess? Absolutely not, however I wouldn't be surprised if there were in fact multiple possibilities for the afterlife.
In the case of having similar roles, I think it could very well be an expression of a different God in a different culture, however I think that you have to examine that God or Goddess in their entirety and really meditate on whether or not they are truly the same entity and not make snap judgements by going "Hearth Goddess, Hearth Goddess, definitely equal". You must examine the God or Goddess fully and then really decide if it is the same as a God or Goddess within your own pantheon. After all, just because I am a student doesn't mean that all other students are me. Likewise, we know each doctor is an individual. Dr. Martinez is not Dr. Li simply because they have the same role in life.
So then, that comes to the question of how many Gods and Goddesses exist. I have heard some Hellenists state 12, 24, or 36 with all other Gods being aspects of those. Some Hindus state that there is only one God with all other Gods being aspects of that one God. Many Shintoists would state that there are 8 million gods(kami), if you translate the phrase literally at least. Colloquially it means that they consider there to be a practically infinite number of Gods(kami).
How then do you decide who is right? Most people simply go with whatever their particular faith says or model the number off their faith's 'central' Gods. Though many faiths don't require the adherent to 'come up with' a number. That is something that is completely and totally going to vary from person to person. Me? I'm going to just say I flat out don't know, and am only equating when I just feel really strongly about it. So far I've only equated Neith-Athene-Saraswati and Hanuman-Hermes-Mercury and Sol-Helios. That's it. I used to equate Freyja and Aphrodite but I've since pulled that back for further examination. Regardless of the equating I have made though, I won't begin to worship Athena in an Egyptian or Hindu fashion. I will treat them separately and as if they are all distinct individual goddesses, because at the end of the day I'm just incapable of being one hundred percent certain. I'd hate to rob a god or goddess of what they rightly deserve because I've made a mistake and confused them for some other deity.
Our ancestors used equating the gods as a means to understand foreign gods and cultures. I'm also sure that there must have been some dissonance in finding out that someone else also believed that their gods and goddesses had absolute control over the universe. Instead of taking the route of 'shared power' they took the route of 'same gods, different names'. I personally suspect that the first answer is right for some gods and the second is right for others. Which is which for which particular god and goddess is up for us to deduce on our own.
I guess if you could take anything from this it would be two simple things. Treat each God and Goddess as if they are distinct and equate them with care.
|Athena by nattywoohoo on Flickr|
|Neith and Amun by risotto al caviale on Flickr|
|Saraswati by Delphine Wa-Dag on Flickr|