"too" is informal and has other meanings and can be confusing because there are other words with other meanings that sound exactly the same when spoken. "as well" is more formal and easier to understand in more formal conversations. "me too" is universal and instantly understood. If you told me that you have a cell phone, I would not answer "me as well". "as well" is not an idiom. "as" means the same, equal to. "well" means to the extent of, limit or measure of. "She does not sing as well as her mother". Both "as" and "well" have other meanings.
Good Q!