Given that he is using the present form of the verb to be, should no be using make instead of made?
No: made is correct. The passive voice is constructed from be plus the past participle. (In the case of to make, the form made serves as both past participle and past tense — and this is true of most English verbs — but you can see the difference with a verb like to see: we say "I am seen", not *"I am see" or *"I am saw".)
[…] and in this construction I don't see the "person".
The "person" is the object of make. Your example uses the passive voice, so the object is converted to a subject (namely "I"). "I am made to shut up" means "[Someone] makes me shut up."