Κοινωνικοποιώ το κατοικίδιο.

Because Greek often leaves out subject pronouns when the verb already shows who the subject is.

Here, κοινωνικοποιώ means I socialize / I am socializing, so the I is built into the verb ending.
If you wanted to add I for emphasis, you could say:

Εγώ κοινωνικοποιώ το κατοικίδιο.

But in normal speech, Εγώ is usually unnecessary.

It is the 1st person singular present tense form of the verb κοινωνικοποιώ.

So it means:

• I socialize
• I am socializing

Greek present tense can cover both the simple present and the present continuous, depending on context.

Το is the definite article meaning the, and κατοικίδιο is a neuter singular noun here.

So:

• το = the for a neuter singular noun
• κατοικίδιο = pet

Greek nouns have grammatical gender, and κατοικίδιο is neuter, so it takes το.

Yes, here it is being used as a noun.

Originally, κατοικίδιο comes from an adjective meaning something like domestic / kept at home, but in modern Greek it is very commonly used on its own to mean pet.

So το κατοικίδιο is a normal way to say the pet.

It actually is the object, but with many neuter singular nouns in Greek, the nominative and accusative forms are the same.

In this sentence:

• κοινωνικοποιώ = the verb
• το κατοικίδιο = the direct object

Even though it is in the accusative, it still looks like το κατοικίδιο.

It can mean either one.

The Greek present tense often covers both:

• I socialize the pet
• I am socializing the pet

Context tells you which one is meant.
If someone is talking about a routine, it may mean I socialize the pet.
If they are describing what they are doing right now, it may mean I am socializing the pet.

Yes, it is understandable and natural, especially in contexts like pet training, behavior, or animal care.

Κοινωνικοποιώ is a real and useful verb for socializing an animal.
That said, in very everyday speech, people might also use longer expressions such as:

• Το συνηθίζω σε ανθρώπους. = I get it used to people.
• Το μαθαίνω να είναι κοινωνικό. = I teach it to be social.

So Κοινωνικοποιώ το κατοικίδιο is correct and natural, but it can sound a bit more formal or instructional depending on context.

You would say:

Κοινωνικοποιώ το κατοικίδιό μου.

That means I socialize my pet.

Notice the form:

• το κατοικίδιο = the pet
• το κατοικίδιό μου = my pet

The extra accent in κατοικίδιό appears because of how Greek writes stress when adding μου.

Yes. Greek word order is more flexible than English word order.

The neutral, standard order here is:

Κοινωνικοποιώ το κατοικίδιο.

But other orders are possible for emphasis, such as:

Το κατοικίδιο κοινωνικοποιώ.

That sounds more marked, as if stressing the pet specifically.
So the original order is the most neutral and natural one for a basic statement.

The most important thing for a learner is the stress: it falls on the last syllable:

κοινωνικοποι-Ω

A rough English-style guide is:

kee-no-nee-ko-pee-O

A few helpful points:

• οι is pronounced like ee
• κ is like k
• ω here sounds like o
• the written accent shows you where the stress goes

So the ending -ποιώ is stressed on -ώ.

The plural is:

τα κατοικίδια = the pets

So:

• το κατοικίδιο = the pet
• τα κατοικίδια = the pets

And the whole sentence would become:

Κοινωνικοποιώ τα κατοικίδια. = I socialize the pets.