It is intentional and is emphasized in the light novel. As representatives of First High to the Nine Schools Competition, all participants wore school uniform with the school's emblem to the competition.
From Volume 3 - Chapter 3, right before the ceremony to send off the representatives to the Nine Schools Competition, Tatsuya was handed the uniform for the technician team, which was to be worn during the ceremony and for the full length of the Nine Schools Competition:
"This is the uniform for the technician team. Please put that on during the ceremony in place of the actual uniform."
The one who answered was Mayumi.
Tatsuya was largely aware of why his sister was in such a fantastic mood.
Her happiness was likely caused by the school emblem embroidered over the jacket's left breast.
The emblem was styled after a flower with eight petals.
The emblem of First High.
And not a replacement either, but the symbol of a Course 1 student.
The intramural competition uniforms were largely the same as the normal ones, which was natural since it was only intended to identify which school the contestants belonged to.