Thanks for this new, helpful info! In this case that is the Health status bar. I suggest also another / additional one: Production Progress status bar (maybe in black & white). For finished buildings the progress of what they are producing: improvements like basket, iron ax or sheep, Hoplites ... Also units have this one, the progress of their food, wood etc. harvest. Could use shiftTAB (if unused).
Yes, Imarok is right, I mean a window, which might be kept open all the time (in windowed mode, not FullScreen). It could consist of columns for units and associated details and should be sortable by every column, which means it can be sorted by type giving an easy overview how many units of any type you have (maybe subcriterium: invisible production timestamp or a visible generic unique number, eg. Hoplite #12), or sorted by current task (idle, garrisoned/healing, moving to somewhere, (fighting?), producing food, wood, stone, metal) giving an easy overview how many units are doing what ... Sorted by health (relative to standard value) could be used for example to select the most injured (and move to next temple or so).
Sorted by location could use own, neutral or enemy territory and additionally the sphere of the CivicCenter ("district") in which they are currently to show units sorted together, which are on the map "together". Another possibility would be the option to place a flag or Ankh symbol on the map and to sort units by the (concentric) distance from this symbol to select easily the nearest units (absolute or relative to their speed) ...
In general the units should be selectable like files in the filemanager (drag and modify with shift / ctrl). Also clicking on a unit could simply center the camera on it.
Selections in this list could be also used to group units (instead of picking them on the map).