Nintendo started experimenting with microphone controls back in 1983 with their Family Computer. The second player controller (the Famicom comes with two controllers hardwired to the console) lacked the Start and Select buttons but featured a microphone. It had no capability to differentiate between spoken words, it simply detected the presence or absence of sound. Their western audience would never get to experience it as this option was not available on the NES.
What was it used for? In some cases it would simply reveal some hidden easter eggs or it was used for cheats. In other titles, the use of the microphone was actually required to advance through the game. Finally, it had some purpose in two players games.
This feature supported only a limited number of games from the extensive (1000+ titles) Famicom library. It's hard to compile a full list of supported titles, but these are some of the games I found.
Image credits: Adam Rufino on Behance
Apple Town Monogatari. Easter eggs. SOURCE
Atlantis No Nazo. After getting the microphone power-up, speaking into the microphone freezes all enemies on the screen. SOURCE
Battle Stadium: Senbatsu Pro Yakyuu. Second player time-out. SOURCE
Bokosuka Wars. Higher stats / fight without losing. SOURCE
Doremon. Shout in the microphone to kill enemies, and possibly other functions. SOURCE
Dragon Ball 3: Gokuden. Easter egg in ending sequence. SOURCE
Gambler Jiko Chuushin Ha 2. Character selection options. SOURCE
Ganbare Goemon 2. Use the microphone to cheat the fortune teller into giving you a second fortune. You can also use the microphone to wake up the sleeping guys you encounter in the game. SOURCE
Golgo 13 - Dai 1 Shou - Kamigami no Tasogare. Karaoke minigame. SOURCE
Hanjuku Eiyuu. Cheats. SOURCE
Hokuto no Ken. Reverse Player 2's controls. SOURCE
Kid Niki: Radical Ninja. Access a bonus area. SOURCE
The Legend of Zelda. The game's manual states that Pols Voices hate loud noises. This is a reference to the Famicom version of the game, where they can only be defeated by talking into the second controller's microphone. SOURCE
Kamen Rider Club: Gekitotsu Shocker Land. Increase chance of winning during roulette game. SOURCE
Kid Icarus. Speak into the microphone to haggle with the shopkeeper. SOURCE
Kieta Princess. Speak to a character. SOURCE
Nagagutsu o Haita Neko: Sekai Isshū 80 Nichi Dai Bōken. Reverse Player 2's controls. SOURCE
Nangoku Shirei!!: Spy Vs Spy. Freeze the computer. SOURCE
Raid on Bungeling Bay. In the 2-players mode of the game, shouting into the microphone calls for an allied jet to assist you in battle. SOURCE
SD Gundam Gaiden: Knight Gundam Monogatari. Show correct route in cave. SOURCE
Star Luster. Hidden ship repair. SOURCE
Star Soldier. Get power-up. SOURCE
Super Chinese 2. Get an item. SOURCE
Super Dodge Ball. Easter eggs. SOURCE
Tag Team Wrestling. Get an ally to fight your opponent. SOURCE
Takeshi no Chousenjou. Completion of the game requires the use of the microphone in more than one occasion, for example: showing a treasure map on the game screen, singing karaoke and while playing pachinko. SOURCE
Tashiro Masashi no Princess ga Ippai. Extra 1,800 health. SOURCE
Zombie Hunter. Change title screen colors. SOURCE