Hi Angeliki,
The student trial software code will allow the download of ArcGIS Pro and ArcMap desktop software. The student software code will only authorize ArcMap for 1 year. However, to use ArcGIS Pro, it is using a named user license. It is not included in the Student Trial media. The user_name needs to be created at the ArcGIS Online organization managed by your university. I would suggest you to contact your campus software central and ask for a user-name with an access to ArcGIS Pro (provisioned with an ArcGIS Pro license).
Another option is, for the time being, is to obtain ArcGIS Pro license via Learn ArcGIS program. To support our education community during the COVID-19 outbreak, we offer free access to ArcGIS (including ArcGIS Pro) until August 31st, 2020, via Learn ArcGIS Program. If you are interested in that program, you will need to create a new name_user account at this link: https://learn.arcgis.com/en/become-a-member/. You will automatically receive a name-user provisioned with ArcGIS Pro license. Use the name_user and login created by this program to sign in to ArcGIS Pro that you have installed in your machine. Check FAQs for more information about this program.
Hope this helps.