In my experience, regardless of what RPG system you use (at least almost, I'm sure there are exceptions), you're going to run into situations where the game's reality is fundamentally different from ours in some way.
One of the great things about RPGs is they can challenge our worldviews and put us in situations we might never get to experience otherwise, allowing us to explore worlds full of "what ifs." If your player isn't interested in exploring those kinds of situations, then maybe RPGs aren't their kind of game, and that's OK!
There could be many reasons why your player is unable or unwilling to suspend their disbelief and explore this kind of alternate reality, and the only way to understand where they're coming from is to have an open and honest discussion with your player.
You mention polytheism as the primary concern, here, but there are other things to consider, too. There may be additional conflict regarding topics like the undead, resurrection, demons, devils, or even the idea of magic and casting spells (there is/was much discussion about this sort of thing with the Harry Potter series, for example). These are all essential topics to bring up in conversation with this player and see how they feel about these ideas before they're potentially thrust into a situation they're uncomfortable with.
I haven't encountered this explicit situation where a religious player is unwilling to adapt, but I've experienced plenty of other scenarios where people were resistant to the unknown, the unfamiliar, or something they believed was "wrong" for various reasons. What's worked for me in the past is to try to understand where they're coming from while looking for ways to draw connections between their viewpoints and the scenario at hand. You should definitely be willing to be flexible, but there's only so much flexibility you can afford before you start making the kinds of large changes you've mentioned you want to avoid.
Some immediate examples of things to draw on include classic fantasy series like C.S. Lewis's The Chronicles of Narnia or J.R.R. Tolkien's The Lord of the Rings. Both of those authors had strong religious beliefs yet were able to translate them to fantasy settings that in turn played a part in inspiring D&D. Look for common ground in your conversation with this player. Find what they do or don't like about fantasy, and see where you can meet in the middle. Then have a talk with all the players together and set expectations. This kind of conversation has worked for me in the past - setting expectations early and having one-on-one discussions about issues of concern goes a long way. I have pre-empted issues many times - not just in RPGs - by speaking with an individual or a group beforehand about expectations. When our expectations were in conflict, sometimes we were able to discuss and reach a compromise or a better understanding of the situation. Other times there was not much willingness to compromise, and that’s when you have to make the unfortunate call of whether it’s worth it to try to push through, or just cut the individual loose. The important thing is not to single this player out and say things like "how could you possibly think that" and "don't you know it's just pretend," etc. Ridicule isn't a great way to discuss things in general, but it's a particularly poor choice when someone is already having a strong emotional reaction to the topic at hand.
The long and short of it is, there's not a great way to "convince" your player to play D&D, nor would I particularly recommend pushing too hard if this player decides it isn't for them. However, there's also not a great way to sculpt D&D to strictly conform to their ideals without potentially making some big changes to the setting or perhaps even certain game mechanics. It's all about perspective, and hopefully your player can come around to the idea that we play fantasy characters in a fantasy world that is fundamentally different than ours.
If not, then the answer may be that you just don't play D&D with them.
The common wisdom is "no gaming is better than bad gaming," and I've seen this especially hold true for new players in RPGs. If it's a negative experience for someone, regardless of reason, that person is unlikely to want to try again. Remember to consider the experience for everyone at the table.
If it can understand morality enough to have an alignment, it can worship whatever deity it chooses to
This is a bit of a strange situation; in comments, you mention how its stats are as per the Monster Manual, so an Intelligence score of 3, but it is also Lawful Good.
RAW, generally beasts with an Intelligence of 3 or less are unaligned, which makes sense because they wouldn't be smart enough to understand morality. It also makes sense to me that they wouldn't be able to comprehend gods either, so dogs wouldn't worship anything.
It sounds like you have misunderstood alignment here, which are to do with morals and principles, I'm not sure that "pack animal instincts" can be thought of as "Lawful Good". The dog isn't moral because people call it a "good dog", and obeying their "master" isn't lawful as that's just their pack mentality, not a choice the dog is making based on its "principles".
However, if, despite that, you decide that this dog is somehow Lawful Good, this implies that, despite its low intelligence, it can grasp the concept of morality, so if you rule that it also understands gods, then at the very least, this dog may worship whatever god you feel makes sense for that character.
They start with their own personality, yet they can and do change afterwards.
This question is rather difficult to answer in a manner that would satisfy everyone: it is best left to individual interpretations based on precedents. I suggest you have a look at the list of deities who were previously mortals and pick some examples and read about them on the FR wiki.
If you want to focus on a couple of well-documented examples, I would suggest you read on Mystra, who has had at least three incarnations. The mortals who were chosen to carry the mantle by each previous incarnation carried some of their own personality with them, as evidenced by the change in alignment.
However it is also true that the personality can and does change to adapt to the job. Kelemvor, the mortal lover of Midnight who became Mystra, was an unusual god of the dead right after his ascension. For example, he was "lenient" on the faithless and the false. Yet this created imbalance in the world and as time went on he readjusted himself for his job. Quoting from the FR wiki article:
Kelemvor and Midnight's ascension and the ensuing changes in personality are described in the Avatar Series of novels, in particular the last novel of the series Crucible: The Trial of Cyric the Mad. Another ascended deity whose story is described in detail is Finder Wyvernspur - his mortal life is recounted in the Finder's Stone trilogy, while he is a deity in the novels Finder's Bane and Tymora's Luck.