Meu vestido é igual ao seu.

Because meu has to agree with vestido, and vestido is a masculine singular noun.

In Portuguese, possessives agree with the thing owned, not with the gender of the owner.

So:

• meu vestido = my dress
• minha saia = my skirt

Even if the speaker is a woman, she still says meu vestido, because vestido is masculine.

Because grammatical gender in Portuguese does not always match real-world ideas.

A dress may be associated with women, but the word vestido is still grammatically masculine. That is why it goes with:

• meu vestido
• o vestido
• seu vestido

This is something learners just have to memorize with the noun.

É is the third-person singular form of the verb ser (to be).

So:

• Meu vestido é... = My dress is...

It is used here to make an identification/comparison statement.

Igual means equal / the same / identical, depending on context.

In this sentence, it means that one dress is the same as another in some relevant way.

A very important point is that igual is normally followed by a:

• igual a = equal to / the same as

So the structure is:

• Meu vestido é igual a...
• My dress is the same as...

Because igual normally requires the preposition a.

So the full structure is:

• igual a + o seu

And a + o contracts to ao:

• a + o = ao

That is why you get:

• igual ao seu

This literally works like equal to yours.

Because seu refers back to an implied masculine noun: vestido.

The full idea is:

• Meu vestido é igual ao seu vestido.

But Portuguese often omits the repeated noun when it is obvious. Since vestido is masculine singular, the possessive also stays masculine singular:

• o seu = yours (your dress)

If the hidden noun were feminine, it would change:

• Minha saia é igual à sua.

No. Seu / sua can be ambiguous in Brazilian Portuguese.

Depending on context, seu can mean:

• your
• his
• her

In many situations, Brazilians prefer clearer alternatives such as:

• dele = his
• dela = hers

So while Meu vestido é igual ao seu can mean My dress is the same as yours, in another context seu could be interpreted differently unless the situation makes it clear.

Because Portuguese often leaves out a repeated noun when it is already understood.

So these mean essentially the same thing:

• Meu vestido é igual ao seu.
• Meu vestido é igual ao seu vestido.

The shorter version sounds natural because vestido is already clear from the first part of the sentence.

Yes, you could, if you are using tu.

• teu / tua go with tu
• seu / sua often go with você

So:

• Meu vestido é igual ao seu.
• Meu vestido é igual ao teu.

Both can mean My dress is the same as yours, but which one sounds natural depends on the region and on how people normally speak there.

In much of Brazil, você and seu are very common.

Yes. Both are possible.

You may hear:

• Meu vestido é igual ao seu.
• O meu vestido é igual ao seu.

Brazilian Portuguese often uses the definite article with possessives, but leaving it out is also very common, especially in simple sentences like this.

So both sound natural, though usage can vary by region and style.

Not exactly.

In comparisons like this, igual is the normal word:

• Meu vestido é igual ao seu.

Mesmo often appears in expressions like:

• o mesmo vestido = the same dress
• a mesma cor = the same color

So compare:

• Meu vestido é igual ao seu. = My dress is like / the same as yours.
• Estamos usando o mesmo vestido. = We are wearing the same dress.

They are related in meaning, but they are not always interchangeable.

The possessives and the contraction would change to match the feminine noun.

For example, with saia (skirt):

• Minha saia é igual à sua.

Why?

• saia is feminine
• so you use minha and sua
• a + a contracts to à

So:

• igual a + a sua → igual à sua

This is the feminine parallel to:

• Meu vestido é igual ao seu.