I have three wallets. Each wallet has the same shape, but a different color. The wallet has a zippered pocket on one side.
Does "The wallet" in the sentence above I created make sense? I don't like to repeat "each".
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You can't use "the" since it is not clear "which wallet" you mean. I know that you mean that they all have a pocket, so that is what you have to say.
You don't need to use "each" in the first part either. You can use the plural since you are talking about all three wallets:
... The wallets all have the same shape but different colours.
... All the wallets have a zippered pocket on the side.
You could equally say "... Each wallet has a zippered pocket..."
If you aren't talking about one particular wallet that has a zippered pocket (as opposed to the other two which do not), then using the here is awkward and confusing.
If you mean to refer to all of the wallets, then the following is the way of doing so that makes the least amount of change to your sentence:
I have three wallets. Each wallet has the same shape, but a different color, and
the wallet hasa zippered pocket on one side.
Or without the annotation:
I have three wallets. Each wallet has the same shape, but a different color, and a zippered pocket on one side.
The phrase inside the pair of commas will be taken as nonrestrictive information, so what follows the second comma is interpreted as applying to what precedes the first comma, with the information inside the pair of commas being parenthetical:
I have three wallets. Each wallet has the same shape (but a different color) and a zippered pocket on one side.