My crew is wrapping up our very first campaign together (my first one ever, Curse of Strahd) and are starting to plan new characters. I told everyone a little bit ago my plan was to go with a goliath totem barbarian/circle of the moon multiclass for next campaign. Another person recently said they were going to play a spore druid, then said they would be multiclassing into totem barbarian, and now is talking about switching the druid to circle of the moon druid. Our first campaign everyone played pretty unique characters and there wasn't a whole lot of overlap although there was some multiclassing into classes other people were playing, but we didn't have any of the same classes. Is it normal for 2 people to play such similar builds with the same multiclasses/subclasses? I feel like we're gonna get in each others way a lot if I don't change my character.

No, this is not normal.  I would recommend you talk with the other player - explain that you expressed the idea on (provide a specific time/event that pinpoints when you came up with the idea), and that you have been excited to play the character for however long it has been. You should also express your concern that they are making a redundant character to what you already said you would be playing, and that might cause the party to have issues in terms of diversification of threat and challenge responses. 

If they do not change, elevate your concerns to the DM, explaining all of the above. 

Then you should make the decision if you want to change your character (or not play with people who are willing to let this kind of thing slide over objection of the person who had the idea first), but exhaust talking with the other player and the DM before you go changing your own plans.

I agree with Caerwyn. It is really weird, and to use an old middle school phrase, you called it first. They should not do it.

If you are left with no other choice, and you end up with two of them, then the only suggestion I'd have would be to kind of lean into it, and see if you can play twins or something. The role play might make it more palatable. 

It takes two rounds to set up raging+wild shape.  A lot of fights in 5E are over by then.  Also, Rage and Wild Shape are both things that a character can do a finite number of times per day.  If they can use them in every fight, do more fights per long rest.

To echo the others, no, it isn't normal to have two of the exact same build in a single party. Definitely have a talk with the player and your DM. It's also possible the other person didn't remember you mentioned calling dibs on the concept, or was secretly hoping you'd come up with something else. There's no harm in communicating that you had that character prepared already and aren't comfortable with a redundancy. 

Also, congrats on making it through your first campaign!

My advice is if the two of you go with similar characters like that, write your back stories together so that your two characters are members of the same tribe and are friends. 

I’m playing in a pick up campaign where we did something like that, although we’re playing different races and classes. Every PC in the campaign is from the same area and we’re all childhood friends and we have one goal in common. 

First go to the other player, it could be a misunderstanding or they could just change when gently confronted about it.

If they do not change or refuse to, and have no reasonable explanation for having that similar of a character to you, then as other's have said, take it to your DM. He is the boss and may be able to solve things.

However, you should do your best to not hurt the other players feelings when explaning your situation to them, that could make the whole thing a much bigger deal than it already is. An angry player is a lot harder to reason with then a slightly frustrated one.

Quote from Tim >>

My advice is if the two of you go with similar characters like that, write your back stories together so that your two characters are members of the same tribe and are friends. 

I’m playing in a pick up campaign where we did something like that, although we’re playing different races and classes. Every PC in the campaign is from the same area and we’re all childhood friends and we have one goal in common. 

 There's one thug having similar backgrounds...same classes and subclasses? Yeugh. You end up sharing the same niche, which means that you're sharing the same limelight, and that'd not good. Especially if you're doing 4d6k3 where one will likely over power the other and put them in permanent shadow.

Sharing classes (or even different classes but occupying the same niche) shouldn't ever really be done, unless it's a gimmick (eg a gaggle of Wizards going on an adventure), but even then...at least pick different subclasses.

One of them needs to change their build and keep this concept for another campaign. Ideally, it would be the one who picked it second, but one of them needs to change it.

Nothing wrong with playing the same classes/subclasses, there are plenty of other ways to distinguish yourselves during play, I wouldn't worry about it.  Like someone else said up above have fun with it, maybe you're from the same school/teacher or something.

I would personally probably just pick a different character, but I have a wide range of character that I enjoy playing. It would be worth considering what roles you each served in your previous campaign (melee, face, ranged damage, etc.). If you played something similar before, you may want to consider backing off of this character to let the other person step in this time, but if instead they played something similar and it's a character that you really want to play then maybe it would be worth mentioned it to them or the DM.