Gameboy Advance has 160 vertical lines. In HD60 mode (what GBI HF-OSSC uses) this is doubled internally to 320 in a 360 height frame. Then using 384p x3 mode on OSSC this is tripled to 960 in a 1080 height frame (i.e. total scaling is x6). So there is naturally always black bars on top/bottom (and sides), preserving original aspect (3:2) with maximum integer scale.
However, if you are running GB/GBC games, it has only 144 vertical lines. So end result is 6×144 = 864 vertical lines.
If you absolutely wish the game area to entirely fill the vertical space, you can try to reduce V.Active area on the OSSC so that your display will scale the last bit. But this may not be compatible with your display.