The Photoshop measurement log can be found in the window menu. Open it and by default, nothing will be in it.
The panel includes a single button for 'record measurements' and that only activates when you add something to your document that it can use such as a selection, a ruler or a count. It will become active, if you want to save the data to the log then click it. The right side menu allows you to change the scale used as well as the items recorded via the 'select data points'. You can also record them via the right side menu as well as via a shortcut. The menu also allows you to export the current selected items and save the data to a text file which can then be imported into Numbers or Excel etc. The right side menu also allows you to select all the entries as well as delete the selected entries. You can see as you add data, that the panel includes the date, file name as well as angle and scale and length etc depending on the tool being recorded.
You can create a selection using a variety of selection tools and the feature will record the number of selections, the perimeter, the circularity, the area, the gray scale minimum and maximum and more. It records the sum of all the selections as well as the individual selections
The ruler tool only adds one ruler at a time (wish it would add more) but records the file, date etc as well as the angle and length, so super useful for recording useful distance / spacing information. Weirdly, it does not record the origin or end points. Note that you have to click the record button each and every time if you want to add an entry to the log. You can move the ruler using the ruler tool and then record it again.
The count tool is recorded but it only records the number of counts added and not the positions etc https://youtu.be/sISXITYtwD0
The menu offers much the same as the right side menu but it does offer one additional option and that place scale marker where a type / layer is added in a group indicating the scale used. It also offers quick access to the ruler tool and count tool (but weirdly doesn't offer selections)