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I'm confused about the use of the wo/o particle. I've read many times that is used to indicate direct objects, but in phrases like this there is a direct object and there isn't a particle wo to indicate it.
Watashi wa Hana desu.
At least in Spanish (not sure in English) "Hana" would be considered the direct object. It is wo omitted in certain phrases or the meaning of "direct object" is not exactly the same as in other languages or how it is?
Not all complements are direct objects.
Let's look at each language, one by one:
I am Hana.
Be is an intransitive copular verb, here in its form am. It takes Hana as a copular complement, but that complement is not a direct object. Instead, it's what is traditionally known as a "subject complement", sometimes called a "predicative complement" in modern grammar.
Why? Because the grammar differs from that of a direct object. You can have an adjective as a copular complement, but not as direct objects.
(Yo) soy Hana.
Ser is a copular verb, not a transitive verb, here in its form soy. It takes Hana as a copular complement, but that complement is not a direct object. Almost all uses of Spanish verbs can be considered transitive or intransitive, but in this use ser is usually put in a separate, very small group of copular verbs with their own special grammar.
Why? How does a copular complement differ from a direct object in Spanish? Well, try putting a pronoun in that position. Object pronouns take accusative case, but as copular complements, pronouns take nominative case instead. So copular complements can't be considered a kind of object.
(Watashi wa) Hana desu.
Desu is a copular auxiliary, but not a verb. In this use, it directly follows its copular complement, Hana, without any particle in between. This complement is not a direct object. As you've noticed, Hana cannot be marked by o here, which is some of the best evidence we have that it's not a direct object.
You'll learn later that sometimes o marks things other than direct objects, but direct objects can pretty much always be marked by o. This is an "accusative case particle", the equivalent of using the object form of a pronoun in Spanish.
As you can see, the grammatical details in each language differ, but Hana isn't a direct object in any of them.
Note: I've transcribed を as o here, but it's the same thing you wrote as wo.
