Yes, the verb here is "zich herinneren" (not just "herinneren"). "Zich herinneren" is an examp[le of what is known as a reflexive verb. There are quite a few of them in Dutch, and as an English speaker you probably just have to memorize them.
The verb "zich herinneren" is a ditransitive verb. That is, it takes two direct objects: the reflexive pronoun plus whatever it is that is being remembered.
Note that "zich" is the reflexive pronoun used in the third person singular and third person plural. In the first person singular, the reflexive pronoun is "me". That is what Duo is using here.
Note that the reflexive pronoun always "reflects" the subject of the sentence:
ik -> me
jij -> je
hij/zij -> zich
wij -> ons
jullie -> je
zij -> zich
That really depends on the sentence. 'Niet' often slightly changes the meaning of a sentence.
"Ik loop niet elke dag naar de Albert Heijn (supermarket)" This means that you sometimes go to Albert Heijn, but not everyday. "Ik loop elke dag niet naar de Albert Heijn" Now this sentence would be incorrect, though if you add a contradistinction it is: "Ik loop elke dag niet naar de Albert Heijn, maar naar de Jumbo.
"Ik kan me elke dag zijn naam niet herinneren" This means that everyday you can't remember his name. "Ik kan me niet elke dag zijn naam herinneren" Now this means, that it is not everyday that you can remember his name. Some days you can, some days you can't.