Learn Finnish: Ihmisellä on oikeus puhua. - A person has the right to speak.

Finnish does not have a direct verb for "to have". Instead, it uses the adessive case (meaning "on" or "at") plus the verb on (is). So, ihmisellä on literally means "at a person is", which translates as "a person has".

This lesson is about the passive voice, so why isn't there a passive verb in this sentence?

That is a great observation! There is no passive verb here. However, using ihminen (a person) is another common way to make general statements about what "one" or "people" do. In Finnish, you will often choose between the passive voice and the "a person does" structure to express these general ideas.

Could we use the passive form puhutaan (is spoken) after oikeus?

No, because the noun oikeus (right) requires the basic dictionary form of a verb (the first infinitive) to explain what the right is for. Therefore, it must be oikeus puhua (the right to speak).

If this means "people have the right", why is ihmisellä singular instead of plural?

In Finnish, it is very common to use the singular form ihminen to represent humanity or people in general when talking about universal rules or rights. You could say ihmisillä on (people have), but the singular version is often preferred for official-sounding general statements.