I have the same problem on a 5D Mark III running firmware 1.3.5. The HDMI output is always 480i59.94 with absolutely no mirroring despite the preference being set. I purchased a brand new Canon HTC-100 mini-HDMI cable just in case. It's the same 480i59.94 on both a BlackMagic UltraStudio Recorder 3G and a LG WebOS OLED TV.
This may have broken somewhere in the 1.3.x firmware range.
After downgrading to every possible firmware from 1.3.5 to 1.2.1, I found I still couldn't get more than 480i59.94 output with any display or capture card (shown on less advanced displays as 480p). Finally I took a close look at the mini-HDMI port and found that half of one of the DDC data pads is missing, and it's one that's critical for EDID negotiation. This explains why it never escalates to Full HD. I suspect it could also be why mirroring mode never activates for clean HDMI output. For anyone else having this exact problem of no FHD and no mirroring, chances are you need to replace the HDMI C port (mini-HDMI), which is unfortunately surface-mount electronics.
12/05/2023: New firmware updates are available.
09/26/2023: New firmware updates are available.
08/18/2023: Canon EOS R5 C training series is released.
07/31/2023: New firmware updates are available.
05/18/2023: New firmware updates are available.
03/30/2023: New firmware updates are available.