he likes playing 怎么也是对的

like to do 强调一次性
like doing 强调习惯性
如：I like singing, but I don't like to sing for you

可以这么说，但是意思有一些不同。

playing 分词表示 经常性的
to do 表示某一次

he likes playing 表示他平时的爱好就是玩
he likes to play 表示他这次想玩

he likes playing football usually, but he likes to play basketball these days.
他平时喜欢T足球,最近几天他却喜欢打篮球.