It's really important to separate HDR and SDR in this discussion - as the same titles mastered/graded for the two different dynamic ranges can appear very different, and any HDR->SDR tone mapping (squeezing an HDR source into an SDR signal) conversion will always have side-effects.
Most HDR displays also have different black level, contrast, brightness etc. settings for SDR and HDR10/HLG formats and thus you need to calibrate separately for both - often on an input-by-input basis. (However Dolby Vision content often inhibits/overrides some of these controls)
I've seen no evidence on my Pi4B LibreElec install that AVC/h.264 and HEVC/h.265 Rec 709 HD SDR content is replayed with any difference in black levels, white levels etc. (I master content in both formats in standard video levels (aka 'Limited') from broadcast quality masters, which are by default limited range, for personal use)
There is clearly a difference between the same title mastered in HD Rec 709 SDR h.264 and UHD Rec 2020 HDR10 h.265 - but this is to do with the colour gamut (Rec 709 vs Rec 2020) and the SDR or HDR EOTF (SDR/BT.1886 vs HDR10's PQ ST.2084), not the codec. In some cases where the mastering has taken place with different colourists or the same colourist doing two separate grades, rather than a tone mapped down conversion, different decisions will be taken artistically between the SDR and HDR 'looks'.
It's important to remember that HDR10's PQ EOTF (the bit that makes it 'HDR') explicitly defines an absolute 1:1 mapping between video levels and output light level from the display, whereas the SDR standard used for regular HD (and some non-HDR UHD) content has no such explicit link and is a relative standard. This difference can often make UHD HDR content look 'dim' or 'dark' compared to the same title mastered for HD viewing in SDR.
(PQ = Perceptive Quantisation, Colour Gamut = the definition of what actual colour the red, green and blue primaries are in the real world, EOTF = Electro Optical Transfer Function = The relationship between the video levels in the signal and the output levels you see on a display. Tonemapping is the conversion between a wider colour gamut and/or dynamic range and a narrower gamut and/or dynamic range. It's the process that decides what you throw away and how you make the SDR signal either reflect the original scene or how you make it reflect the experience of viewing the HDR mastering on an SDR display - which are two very different approaches)