Maybe it's because Ice Bear begins with an "I", which means the writers can do funny stuff when Ice Bear's trying to speak in first person. Like in "The Audition" for example. Ice Bear went from saying " I... I..." to "Ice Bear".
I think Ice Bear is called 'Ice Bear' because yes it does have a better ring to it than 'Polar Bear'; and because, like what The 1 and only scaredy cat said, it's just another name for polar bears. ¯\_(ツ)_/¯ They're called 'white bears' too, but... let's just face it. 'Ice Bear' is the best name out of all the options.