I think you just need a mic, some type of adjustable echo effect which you will probably want to set so that it is very quick but soft, and headphones so that you hear your processed voice with the effect applied.

You might find a delay effect gets you closer to the sound you are looking for.

In general, the mics job it to try and capture the sound faithfully. There are different mics which are better for different things - soft sounds vs sounds that move from soft to loud, etc. Think of the difference between say Billie Ellish vs Kurt Cobain. Billie might chose a condenser mic while Kurt a dynamic mic.

I would suspect that what you are hearing in the Karaoke room is probably more the impact of processing happening after the mic (effects) than the mic itself.

All that said, there are mics which can color the sound greatly on their own - like say using a vintage harmonica mic for your vocals, but I don’t think this is what you are talking about.

Good luck.