Learn Polish: Hej, dzisiaj mam nowy telefon. - Hey, I have a new phone today.

In Polish, you usually drop the subject pronoun (like ja for "I") because the verb ending already tells you who is doing the action. The verb form mam specifically means "I have", so you don't need to say ja mam unless you want to emphasize it.

I learned that the verb mieć (to have) requires the accusative case. Why hasn't the word telefon changed its ending?

You are right that mam takes the accusative case (used for the direct object of a sentence). However, telefon is a masculine inanimate noun (a masculine noun that isn't a person or animal). For these types of nouns, the accusative form looks exactly the same as the basic dictionary form.

Why is the adjective nowy used here instead of nowa or nowe?

Adjectives must match the gender of the noun they describe. Because telefon is a masculine noun, we use the masculine form nowy. And just like the noun, the masculine inanimate adjective doesn't change its ending in the accusative case.

Is hej used exactly the same way as "hey" in English?

Yes, it is very similar! Hej is an informal greeting used with friends, family, and peers. You wouldn't use it in formal situations, like greeting a boss or a shop assistant, where dzień dobry is better.