Eu tenho o mesmo problema que você tem.

Yes. In Brazilian Portuguese, subject pronouns are often omitted because the verb form already tells you who the subject is.

• Eu tenho o mesmo problema que você tem.
• Tenho o mesmo problema que você tem.

Both are natural.

Including eu can add:

• emphasis
• contrast
• clarity

For example, if you want to stress I have the same problem, you might keep eu.

Because você uses third-person singular verb forms in Brazilian Portuguese.

So:

• eu tenho = I have
• você tem = you have

Even though você means you, grammatically it behaves like he/she for verb conjugation.

If you use tu, the standard form would be:

• tu tens

But in Brazil, usage of tu varies a lot by region, and many speakers mix tu with third-person verbs in everyday speech.

O mesmo problema means the same problem.

Breakdown:

• o = the
• mesmo = same
• problema = problem

So mesmo here means same, not even or really.

It also has to agree with the noun:

• o mesmo problema = the same problem
• a mesma ideia = the same idea
• os mesmos problemas = the same problems
• as mesmas ideias = the same ideas

When mesmo means same, it normally comes before the noun:

• o mesmo problema
• a mesma coisa

That is the standard pattern.

If mesmo appears in other positions, it can mean something different, such as:

• ele mesmo = himself
• agora mesmo = right now
• mesmo assim = even so

So in this sentence, mesmo is before problema because it means same.

Because not all Portuguese nouns ending in -a are feminine.

Problema is a masculine noun:

• o problema
• um problema

This is a common pattern with some words of Greek origin, such as:

• o sistema
• o tema
• o programa
• o problema

So you say o mesmo problema, not a mesma problema.

Here, que connects problema to the clause você tem.

So:

• o mesmo problema que você tem

literally works like:

• the same problem that you have

In this sentence, que is functioning like that in English.

It introduces extra information about problema:

• which problem?
• the problem that you have

Both are possible, but they are slightly different in feel.

Full version

• Eu tenho o mesmo problema que você tem.

This is fully explicit and very clear.

Shorter version

• Eu tenho o mesmo problema que você.

This is also very natural in conversation. The final tem is understood and often left out.

So:

• with tem = a bit more complete or careful
• without tem = very common in everyday speech

Not really in this sentence.

In English, same often appears with as or a clause like that you have. Portuguese uses que here in a structure that links the noun to the clause:

• o mesmo problema que você tem

A more natural English match is:

• the same problem that you have
• or simply the same problem as you

So even if the overall meaning matches English comparison structures, que here is not literally functioning as than.

In most of Brazil, você is the normal everyday singular word for you. It is usually neither especially formal nor especially informal; it is just standard spoken Brazilian Portuguese.

Other possibilities are:

• tu = used in some regions
• o senhor / a senhora = more formal or respectful

So this sentence sounds like normal everyday Brazilian Portuguese.

Yes. Some common alternatives are:

• Tenho o mesmo problema que você.
• Eu tenho um problema igual ao seu.
• Estou com o mesmo problema que você.

These are all natural, but they are not identical in structure.

Differences

• o mesmo problema que você tem = the same problem that you have
• um problema igual ao seu = a problem equal/similar to yours
• estou com = literally I am with, but often used in Portuguese for temporary conditions or issues

The original sentence is completely correct and natural.

Yes, it is natural and correct.

Eu tenho o mesmo problema que você tem is a normal way to say this in Brazilian Portuguese.

In very everyday speech, many people would shorten it to:

• Tenho o mesmo problema que você
• Eu tenho o mesmo problema que você

But the full sentence with both verbs is absolutely fine.