Why would it be illogical? Is this just a parody of what a Muslim would say, or is there actual reason to think multiple gods might be illogical?

When polytheists say that there are multiple Gods, they sometimes refer to these Gods as having different wills. Now, the Muslim, however, defines God as having the only will. So, there is no scope for possibility of multiple wills. So, the Muslim will outright dismiss the possibility. Hence, my post above. It is not meant as a caricature/parody but as an honest attempt at applying the principle of charity to try and understand where a Muslim is coming from.

Now, when the Muslim charges the Christian or Hindu with polytheism, the Christian / Hindu will defend it by arguing that these different manifestations can simultaneously be called Gods because they are expressions of the same will. Now, how the Muslim will respond, I do not know.

God could make himself known to anyone at any time for any reason.

I have made this argument before against Islam myself. The extension of God -> jebriel -> Mohammed -> humanity at large seems rather tortuous and tenuously brittle. I can only hope an actual Muslim can come to defend this. The argument I have heard is that this life is a test and it would be making this life's test too easy if God himself came down and showed himself to us.