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 ...
You should not omit "the" when talking about a particular zoo.
"The" is sometimes omitted when we are speaking of the "social institution" (a collection of people) and not "the physical building".
You have to go to school until you are 18
We go to church on Sunday.
If you are talking about a particular school then use "the"
I go to the school ...
Not at all. Because 'the' highlights the restrictive or defining function of relative clauses most clearly, examples of this type tend to be discussed to the exclusion of all others.
But there are other cases where relative clauses do not require the definite article. For example:
Any person who is a friend of Bill's is a friend of mine.
A waiter who can'...
Both the windows
Both of the windows
All mean the same thing in most contexts. It all comes down to the use of the definite article "the", which is well documented, but I'll try and explain using this example.
I'm sure you know that you use the definite article when you are referring to a specific thing or a specific set of things. ...
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 ...
Definite noun phrases have different uses in English. A general usage is that the speaker uses a definite noun phrase when they assume that the hearer can identify the referent of the noun phrase.
I like the movie, Pulp Fiction.
The speaker assumes the hearer can identify the referent, that is, the hearer can identity which movie the speaker ...
Is that a rule that I always have to put an article before countable nouns
When the noun is not a proper noun, and referring to something that, in that sentence, can be counted, then yes.
If you can answer the question which X sensibly, then X is countable.
On Horse Market
Proper nouns in English are capitalized. You don't use an article here because ...