What on earth is a fever stick??? I can not guess what it is, if it is not a thermometer and I do not think that a thermometer has any resemblance to a stick!
First, a stick is much longer than a thermometer. Second, a stick something is something you can rely on, which is not physically possible with a thermometer.
Further, you cannot say that a dear child has many names, because I do not think there are many who like to have taken the temperature in the butt ... or even to have a stick up in the butt for that matter ...
The reason for the use of the word "fever stick" can, of course be that he / she cannot remember that it's called a thermometer ... but where is the association to "stick"?