Tongue twisters are sequential poems made up of lines or verses that are constructed with similar consonant sounds (using alliteration). Typically, these are read aloud repeatedly. Now, according to the definition, a poem is any piece of writing in which words are arranged in short lines that are chosen to beautify or rhyme together. So, tongue twisters are considered to be short poems.
As a characteristic of poems, rhyming is an important element. However, rhyming is not a mandatory requirement. Tongue twisters do not always have to rhyme or be meaningful.
The primary purpose of tongue twisters is to tie your tongue into knots, which is also a characteristic of poems. So, tongue twisters are definitely poems. Apart from that, tongue twisters are also a source of fun.
An example of a tongue twister which can also be called a poem:
Peter Piper picked a peck of pickled peppers.
A peck of pickled peppers Peter Piper picked.
If Peter Piper picked a peck of pickled peppers,
Where’s the peck of pickled peppers Peter Piper picked?