How to keep class object during form process
No, if this is a value object then a service not the answer.

I would recommend you look at entities. Drupal will do a ton of work for you if you use (content) entities for your value objects.

See https://www.drupal.org/node/2192175 for example for documentation, also check out the entity module for additional functionality that is not yet in Drupal core.

With content entities, there is a standard EntityForm class that contains an entity and allows you to save it.

That said, what you are doing basically works, $this->student will be available, but be aware that on the first form submission, formBuild() is called again, so it will build another object. The same for entities. If you really want to prevent that, you need to store your object in the database, e.g. with the tempstore API.