Hi Alexandru - George Rusu,
What message do you get when you can't login VS2013 with your Student account?
According to the Developer & Designer Tools list in DreamSpark site, I can't find VS2013. So I'm afraid that the student account just can be used in VS2015.
If you want to use VS2013 for free, please download VS2013 Community or Professional version from below link.
https://www.visualstudio.com/en-us/downloads/download-visual-studio-vs.aspx
Then you can create a Microsoft account through the right-top "Sign in" to sign up a new account, which can let you use VS2013 Community or Professional version for free.
Best Regards,
Weiwei