The other day, I heard someone say:
I want it either.
It comes strange to me. I'm usually used to hear/say:
I want it too.
Is it grammatically correct to use either of them?
*"I want it either" is certainly incorrect, though "I want either" and "I don't want it either" are both standard and common.
I suspect that the person you heard use "I want it either" was modelling their use on the latter.
Here we're following a previous negative statement with agree in a later sentence, an extension of it's adverbial use in:
Some would object to "I don't want it either" (and even more to the more heavily elided "me either") that neither should be used instead. However, the use is certainly common. (Especially in American English, those who would insist on neither are more often British or Irish, though both opinions are found with all forms of English).
Indeed, ?"I don't want it, too" would here be the confusing case; arguably justifiable, but certainly at least strange.
That there is a common use of either that is similar to the form you heard, suggests that they are mistaking one use for the other, especially if they're a non-native speaker, as you say.
No, they are not interchangeable for two reasons - grammar and meaning.
"I want it too." means that I want it.
If I heard someone say, "I want it either" I would think their usage strange but I would understand it to mean "I don't want it."
So the two have opposite meanings in my opinion.
You will sometimes hear "Me either". This is invariably used to mean "Me neither."
This is quite common in informal speech and means "Neither do I."
By extension we can imagine the following conversation.
This has every appearance of being an over-generalisation of version 1. Probably by a non-native speaker.
Apologies if I have duplicated some of what has already been said by others, including in comments. I hope I've added something and also successfully drawn together other people's explanations.