I want to refer to the text which is inside an HTML element, (you can think it is inside a paragraph), can I say containing text of the element? If no, what should I say?
If yes, what is the structure of this adjective?
First, I'm assuming we are talking about a situation like this:
And, as you may have guessed, you would usually call that the "contents" or "text contents" of the element. For example, JQuery API documentation for .text(), where that function is described as:
Get the combined text contents of each element
Your words "containing text" are confusing because it sounds like they are doing the containing. However, it is the text that is in the container.
An alternative correct phrasing would be "contained text of the element".
 JQuery .text() documentation, http://api.jquery.com/text/