Minha filha gosta de história, mas meu filho prefere geografia.

Because the possessive adjective agrees with the thing being possessed, not with the speaker.

• filha is feminine singular, so you use minha
• filho is masculine singular, so you use meu

So:

• minha filha = my daughter
• meu filho = my son

Even if the speaker were male, they would still say minha filha and meu filho.

They are the feminine and masculine forms:

• filha = daughter
• filho = son

This is a common pattern in Portuguese:

• -a often marks feminine
• -o often marks masculine

So:

• a filha
• o filho

And in the plural:

• filhas = daughters
• filhos = sons / children in a mixed group

Because the verb gostar normally requires the preposition de.

So the structure is:

• gostar de + noun
• gostar de + verb
• gostar de + pronoun

Examples:

• Ela gosta de música.
• Eu gosto de viajar.
• Nós gostamos disso.

So in this sentence, gosta de história is the correct pattern.
gosta história would sound ungrammatical in standard Portuguese.

Because preferir works differently from gostar.

• gostar → needs de
• preferir → usually takes a direct object, with no preposition

So:

• gosta de história
• prefere geografia

If you compare two things, Portuguese often uses:

• preferir X a Y

For example:

• Ele prefere geografia a história.

So prefere geografia is correct, and prefere de geografia is not.

Because in this sentence they are being used in a general sense, as school subjects or fields of study.

Portuguese often omits the article in that situation:

• gostar de história
• estudar geografia
• ensinar matemática

This sounds like talking about the subject in general, not a specific one.

If you make it specific, an article may appear:

• Ela gosta da história do Brasil.
• Ele prefere a geografia humana.

Notice that with gostar de, if there is an article, it combines with de:

• de + a = da
• de + o = do

In Portuguese, school subjects are normally not capitalized.

So you write:

• história
• geografia
• matemática
• português

This is different from English, where words like History or Geography are sometimes capitalized, especially in school contexts.

In Portuguese, they stay lowercase unless they begin a sentence or are part of a title.

They are both in the present tense, third person singular.

Why third person singular? Because the subjects are singular:

• Minha filha → she → gosta
• Meu filho → he → prefere

Basic forms:

gostar

• eu gosto
• você/ele/ela gosta
• nós gostamos

preferir

• eu prefiro
• você/ele/ela prefere
• nós preferimos

So the sentence is talking about what they like/prefer in general, now.

Yes, in a sentence like this, the comma is standard and natural.

Mas means but, and here it joins two full clauses:

• Minha filha gosta de história
• meu filho prefere geografia

Since each part has its own subject and verb, the comma is normally used before mas.

So this punctuation is very typical:

• ..., mas ...

Yes. In many varieties of Portuguese, including much of Brazil, you can use the definite article before a possessive:

• a minha filha
• o meu filho

So this is also possible:

• A minha filha gosta de história, mas o meu filho prefere geografia.

In Brazilian Portuguese, usage varies by region, style, and context.
The version without the article, as in your sentence, is completely natural.

A helpful approximation is:

• filha ≈ FEE-lya
• filho ≈ FEE-lyoo

The important part is lh.
This is a special Portuguese sound, not a normal English l.

It is somewhat similar to the lli sound that some English speakers use in million, though it is not exactly the same.

Also:

• história has the stress on TÓ because of the accent mark: is-TÓ-ria
• geografia is usually pronounced with stress on fi: je-o-gra-FI-a in Brazilian Portuguese

Mas means but.

So in your sentence:

• ..., mas meu filho ... = ..., but my son ...

It is not the same as mais.

• mas = but
• mais = more / most

This is a very common confusion for learners because the words look similar.

Examples:

• Eu quero mais água. = I want more water.
• Eu gosto de história, mas prefiro geografia. = I like history, but I prefer geography.

Yes. You could reverse the two clauses and still have a correct sentence:

• Meu filho prefere geografia, mas minha filha gosta de história.

The meaning stays basically the same, but the emphasis changes slightly depending on which part comes first.

The original version is perfectly natural, and so is the reversed one.