Whether or not you additionally add the Commerce functionality to charge users to register, you can include whatever fields you want in a User Registration form.
You first need to create a group of user fields that includes the fields you require, at Toolset > Custom Fields.
Once you have, if you auto-generate the user form markup then it will include all such fields (and you can omit certain fields from the form if you want, or add non-Types fields that are missing from your group).
These fields will appear on the user profile in the back end. (WordPress doesn't have a front-end user profile page, you would need to create one.)
Just one more thing : you say "WordPress doesn't have a front-end user profile page, you would need to create one".
If I use WooCommerce, the person have a front-end user profile page, no?
Otherwise, I could create a page with Types, is that what you suggest?
Yes, there are account pages that come with WooCommerce, but they are WooCommerce pages that do not know anything about any additional custom fields you add to a user profile (regardless of which tool you use to add them).
The choices would be to either customise the WooCommerce account pages (the WC API allows for this with code, or there are plugins to help, see hidden link for an overview).
Or you can make you own account profile page using Toolset.
Just create a static page which will be the account page, and then insert the content and fields you require. If you use the Fields and Views button to insert such fields you'll be able to specify that the user the fields should be displayed for are the current user, so that when anyone looks at that page they will see their own profile fields.
