Do you know if it is possible to assign a date (whatever the format) as the 4th dimension to a geometry in PostGIS? If so, does it recognize it as a date?
Yes, it's possible to store time in the m-value.
No, it cannot be a date type. It has to be numeric.
I've used unixtime, e.g. in https://anitagraser.com/2017/10/15/movement-data-in-gis-9-trajectory-data-models/. It's all based on PostGIS' temporal support, e.g. https://postgis.net/docs/ST_IsValidTrajectory.html
(You can find more examples in https://anitagraser.com/2018/04/16/movement-data-in-gis-12-why-you-should-be-using-postgis-trajectories/)