The spell doesn't prohibit the gaining of experience, so it doesn't… at least not automatically.
You bring up some examples where it might make sense to not gain XP though: the sub-human intelligence giant bug and the alien-minded elemental. A DM might look at those and decide that the former is incapable of learning, and the latter is capable of learning but their normal mind would be incapable of applying those learning experiences later to their true form's life.
The other two examples (the minotaur and the lamia) might inspire the thought that experience gained is still useful, but not fully applicable to their true form. A DM might think that maybe they should earn XP, but not the full amount…
And maybe a DM might think that gaining XP while effectively being a member of a different class should… have an effect? Maybe it can be handled by multiclassing once returned to true form, or just wiped out, or transfered at a reduced rate…?
A DM could have all kinds of useful thoughts about these situations, which the book doesn't answer for them.
Fortunately this doesn't have to be complicated, because it's a rule that the DM can make such rulings according to what makes sense and is best for their group's game. It's their job within the game to make decisions based on real game contexts that the rules don't or can't foresee. The DM at the table is far better equipped to make choices that are best for their game than a designer far away and years in the past, anyway.
So if you're the DM: decide. If you're not the DM: ask your DM.
I don't think there is anything official regarding "back-up" characters in any WotC manuals or guides.
In this specific instance, I think you guys will just have to homebrew it, and decide what's best depending on your campaign setting, your DM, and your PC group dynamic. Either way, I'll try my best to answer your questions from experience:
If this is the case, though, how should we handle their experience? Should all backup characters gain experience at an equal rate as the player's main character? Should they level up at the same rate, but keep the minimum experience for their level?
I think that back-up characters couldn't possibly gain experience at an equal rate as PCs, because situationally, I don't think they'd gain the same XP in combat/story scenarios. If they did, then they would be as involved in the conflict as the PCs, which for story telling and role playing reasons could not make sense. But if you don't quite care for technicalities, then I suppose they could-- their XP might not be justly deserved though.
I also think that would lead to PCs taking death for granted in game; although many players take their characters and backstories seriously, many players care about the leveling up part of their characters. By giving them characters that are where they left off, it might create an environment where death isn't taken quite as seriously.
Is there a different way to handle this entirely?
I think that you guys generally have a reasonable idea; if your campaign is mostly set on the seas, I can understand why "back-up" characters who are crew members might make more sense than other more fantastical, Deus-Ex Machina-y reasons.
What I would personally do, as a DM, is perhaps set them at a level lower than their current PC was when they died, or set the new PC's level to be the same as the lowest level in the party.
There is also the idea that if you are on a boat, ship, or any type of seafaring vessel, at some point you will have to dock at a port-- to restock supplies, conduct maintenance, gather info, etc, etc. This could be a reasonable way to "recruit" new crew members to help fill in the gap left by deceased characters (and a reasonable way to introduce new PCs).
If you guys are open to using the playable races introduced by Volo's Guide to Monsters, a new PC might be a Triton, who guards the ocean depths and might want to join your quest, especially if it is set so close to home. A campaign set almost exclusively on the seas lends itself to including Tritons as a playable character race. I do warn against using these races so openly, as more impressionable players might want to be a "cool" new race, and then you'll have a party full of rare races-- though, if you know how to work with it, that could be fun too.
Best Answer
Generally speaking, if you reward people for things that are fun, it will gradually become less about the fun and more about the reward.
Personally, when we used XP per person, gaming was more focused on doing something cool with your character, as opposed to having fun and reaching the groups goal.
This is basic psychology, it's how the human mind works.
Now whether you want that is plain opinion. That's something you need to decide for yourself.