It's informal but grammatically correct. I think it would be better style to put the "wee wee wee" within quotation marks, I'd hyphenate them to suggest a continuous squeling, and I'd use "cried" rather than "went" to avoid ambiguity, see below. Thus: And this little piggy cried "Wee-wee-wee," all the way home. This is grammatically the same as And this little piggy squeaked all the way home. My mother played it with me, I played it with my kids, and I now play it with my grandchildren. I don't tickle them, I just wiggle their little toe. The "wee wee wee" part is said creatively--jokingly--with extreme rising and falling pitch. "Wee, wee, weeeeeeee, wee-wee-wee-wee-wee" etc. It is NOT being used this way in this poem, but "wee-wee" CAN be an nursery euphemism for urination. If a toddler says "I need to go wee-wee," you lead him to the bathroom. If you say the piggy "cried wee-wee-wee" there is no ambiguity. If you say the piggy "WANT wee-wee-wee" many people will have a fleeting thought that the piggy might be urinating. I once was in a chorus, we were rehearsing the famous Christmastime song "Jingle Bells," and we got to the lines about a sled falling in a snowbank: "The horse was lean and lank Misfortune seemed his lot He got into a drifted bank And we—we got upsot." The chorus director stopped us, and said "Get out your pencils and make a change. We are going to sing 'and THEN we got upsot.' I don't want to have any 'wee-wee' in this song."

"Go" can sometimes be used to mean "said", but it's a colloquial usage. Still, it's not strange to native English speakers, as the meaning of "go" is not limited to moving somewhere. For an animal's sound, we do use "go"! A dog goes woof, a cat goes meow, a little piggy goes wee wee wee...

Yes, it's the sound of a piglet squealing as it goes home. It should perhaps be written as: And this little piggy went "Wee! Wee! Wee!" all the way home. The rhyme is used as a children's game. The 'piggies' are counted off on the child's toes and the "wee wee wee" part is the sound you make as you tickle the child at the end. My mum used to play this game with me when I was little.

It's the sound of a pig squealing.

This is just associated to a sound. For example, in English there are things called "Onomatopoeia". A few examples of these sounds would be "Beep" "Ahh" "Haha" "Ho ho ho" (This is what Santa says" or even "hissssssss" for the sound of a cat being angry.