The simple answer
This いう, which you could gloss as "call", simply takes two arguments. You can call it a "accusative-quotative construction" (which is just a fancy way of saying that there is both an を-marked thing and a と-marked thing).
The actual answer
To learn more about these so-called accusative-quotative constructions, let us play with this sentence:
tyoko-ha [[hikaitousya-ga kawai-i] to] omot-tei-ru
"Choko thinks that 非回答者 is cute."
Another way to say this is by lifting 非回答者 outside of the downstairs predicate and marking it with を:
tyoko-ha hikaitousya-wo [kawai-i to] omot-tei-ru
"Choko considers 非回答者 to be cute."
Note: I will be alternating "thinks" and "considers" in English because "considers" sometimes allows me to build a sentence with similar syntax to the Japanese in English while "thinks" doesn't always work as smoothly.
Another way to say it is by using a complex predicate 可愛く思う:
tyoko-ha hikaitousya-wo [kawai-ku omot]-tei-ru
"Choko considers 非回答者 cute."
In the previous sentence, 非回答者 is not "lifted" out of a downstairs predicate, because 「ちょこは非回答者が可愛く思っている」 is not grammatical. 非回答者 is simply at the top level.
The same sort of thing also works for other predicates, such as
tyoko-ha [[hikaitousya-ga human da] to] omot-tei-ru
"Choko thinks that 非回答者 is dissatisfied."
tyoko-ha hikaitousya-wo [[human da] to] omot-tei-ru
"Choko considers 非回答者 to be dissatisfied."
You can also kind of get rid of the inner predicate like we did with 可愛いと→可愛く, but it doesn't work out perfectly:
tyoko-ha hikaitousya-wo human-ni omot-tei-ru
"Choko considers 非回答者 dissatisfactory."
Namely, the "experiencer" of 不満 is now Choko, not 非回答者.
However, this does not hold for all semantically similar sentences. For example, 次郎はその仕事が楽だと感じた→次郎はその仕事を楽に感じた works just fine (i.e., they have the same meaning).
This is totally my own interpretation, but I think when this transformation takes place, the "experiencer" of the verb switches to the nominative. Sometimes, this can result in a different interpretation, but sometimes, it results in the same interpretation.
In fact, I think the same happens with 可愛く, it's just "experiencing" 可愛い means thinking that someone is cute -- there is no other way to experience it. (For the few others I've thought of, this type of argument seems to hold for all i-adjectives, but obviously not for all na-adjectives as evidenced by 不満).
Anyways, after taking you on that long linguistic journey, I'm basically going to say that a lot of that isn't really relevant. I just wanted to introduce that there are some verbs which seem to accept を and と simultaneously, and those are the verbs that allow what has historically been described as "raising-to-object" or "exceptional case marking" (which means there are many sentences which you can use to get a feel for how the を-marked thing and と-marked thing interact).
Finally, to answer the actual question, this いう is one such verb that accepts both an を-marked thing and と-marked thing simultaneously:
"(He) says (his) name as Hiiragi."→"(He) calls himself Hiiragi."→"His name is Hiiragi."
"(I/we) call him Hiiragi."→"His name is Hiiragi."
In this case, と is not really compartmentalizing a predicate, but instead acting quotatively. Either way, it's the same thing as we saw before.
Unfortunately, I could not find a reference which exactly discusses things using the perspective I gave, but these are in a very, very close ballpark. They discuss the syntax and semantics of these accusative-quotative constructions in great detail without getting too technical and were very fun for me to read.