How can I measure a saddle to buy something similar? I got on quite well with the stock saddle on my tourer, until recently.  That's not surprising as the plastic has broken.  The bike manufacturer isn't great for spares, and I'm sure I could do better, but how can I get something similar?  Try-before-you-buy from a proper shop isn't currently the option it normally would be, so it looks like I'll have to order  something. Are there features I should be measuring on the old one?  Most manufacturers seem to list very little in the way of dimensions, favouring a description.  There's plenty of information out there about putting the new saddle in the same place, but nothing much about getting one similar to the old one.
Summary:

There is no set of measurements you can use to compare seats between brands.
The best way to get the same feel every time would be to find a seat model you like that has an enduring reputation. You may even want to get a spare, just in case.

Details:
The seat a bike company includes on a new bike is most often the one they could get for the lowest price. It's almost impossible to find a match for these seats.
On more expensive bikes bike companies will specify something that is name brand and can be purchased again from various venders.
In either case if you get a seat that works for you it's almost a miracle. Bicycle seats are like shoes - they really need to be tried on. Some companies offer an easy return policy on seats so you can try out seats at home and return them if they don't work.
The measurement that pops up most frequently is seat width.

Every saddle should either come with a width measurement or can be measured.


Even seats with the exact same measured width will have different curvature, length, padding thickness, padding type, frame, design and flex.
Since there are so many factors in what determines how a seat feels some seat makers have web functionality/methodology asking you questions that will allow them to make suggestions for the correct seat. The Selle SMP &quot;Saddle Finder&quot; is one example. Another is the Saddle Finder for Treks Bontrager brand (not a recommendation) They base their recommendation on things like male/female, age, type of bike, type of cycling, frequency of riding and sit bone width based on pant size.
The information provided by seat makers is based on standards created by the seat maker that might help you pick seats that are similar within one seat maker's model range. It is unlikely that one makers information will help when comparing seats from different makers.
There are some seat makers and models that have been around for a long time.
The best way to get the same feel every time would be to find a seat model you like that has an enduring reputation. You may even want to get a spare, just in case.