Is it possible, with Pods or something else dedicated to Wordpress, to create a tags style field as there is when writing a post in Stackoverflow or Wordpress ?
The goal is to create a form on a page and allow a user to enter their own tags
What you want is a custom taxonomy. Tags is an example of a non-hierarchical taxonomy. Categories is an example of a hierarchical taxonomy. Pods will allow you to easily add a custom taxonomy. Here is documentation on the process: https://github.com/pods-framework/pods-code-library/blob/master/learn/content-types/create-taxonomy.md
You can also add a custom taxonomy manually. I recommend using the taxonomy generator provided by GenerateWP: http://generatewp.com/taxonomy/