The C section behaves the same as the flat fin but instead of 1 line that is a hinge, you have 3 lines that are hinges. The long line is still hinged to the block, but because of the short flanges in the C section, the whole section does not tip over with a zero frequency mode.

I tried using Shared Topology with a block, a base surface and a fin. It gives the same result as just a block and a fin, a hinge is created and modal gives a zero mode.

However, if you put the base surface and fin in one component and hide the block, you can use Shared Topology on the surfaces. Then in Mechanical, use Bonded Contact to connect the base surface to the block, you get a fin that stands up and a non-zero first mode.