I believe it should be possible, with some small caveats for disk usage and/or encoding quality. Most of the cheaper USB HDMI capture devices, such as the ones described here present as a UVC video source, which is natively supported in Linux. The raw stream from this can easily be captured using
ffmpeg, provided the SD card or USB disk attached is fast enough to write the data stream. The command, based on the ffmpeg wiki, would be something like this:
ffmpeg -f v4l2 i /dev/video0 -c:v copy -c:a copy output.mkv
In the case the disk was not fast enough, a transcode may be needed, in which case something like this could be used:
ffmpeg -f v4l2 i /dev/video0 -c:v libx264 -crf 25 -c:a copy output.mkv
These are probably not the ideal encoding settings and likely a bit heavy on the CPU. It seems like there might be better options specific to the Raspberry Pi. An alternative to using ffmpeg might be to use OBS Studio, which might be a bit easier to use with its GUI.
The larger issue is that most cheap HDMI capture devices will only capture 1080p at 30fps (but 720p at 60 fps). Is there a specific reason 60 fps is needed? Video capture devices capable of that start to become more expensive. The only device I'm aware of off the top of my head that would work in a similar fashion would be an Elgato CamLink, which presents as a UVC device. However, it does not have HDMI pass through and so an HDMI splitter would be needed. I'm sure there are other devices out there that are Linux compatible and support 1080p/60 with pass-through. Some quick Googling suggests the Elgato Game Capture HD can do it, but only with some fairly complicated.