@Leonid - While your Miro account profile may be a member of an Education Plan team (which offers high-res exports) your account profile can also be a member of a Free Plan team which does not offer high-res exports. 

Given Miro is disabling the high-res export options with a message to upgrade, the board you are showing in your screenshot must not be residing in the same team as the one your fellow student's board is in.

When you sign into Miro, how many teams is your Miro account profile a member of? 

Finding your plan type

Select the team where your board is and then click on the Team profile settings button:

Here you can see your plan type:

Please help. I want to create my team, but the program prompts me to buy a new plan. Actually I have only created two boards on two different command panels on this account. How do I create my own team? I never got the extended plan for students.

I need my account with my boards, which only I can see.

@Leonid 

I never got the extended plan for students.

Are you trying to create a Education Plan team of your own? If so, you cannot do this on you own. You need to apply for one here → https://miro.com/education-whiteboard/2/