Quick question for English native speakers?

It's used frequently and it is completely ok!

It's figurative language, just a better way to illustrate what you mean . It can also be used with other things besides poetry when you want perfect memorization.

You learn a poem by heart, then recite it from memory. But possibly some people say 'recite by heart'.

It is wrong. Recite means ' to say ' or to 'repeat something learnt by heart'. You cannot recite ' by heart '. It should be 'to learn (the poem) by heart and recite'. This is what children do for their class work.

If you mean say the entire poem without having to read the text because you have already memorized it, then it is correct.

It's completely fine and used quite a lot , you should know that!

it does sound strange when you think about it, but almost all phrases do. so don't worry about it.