Hmmm ... you're quite right. You can see the Task name field but not the Project field.
Looks like you'll have to do one of two things:
1. Do the "analyze timescale data" on the individual projects, save the individual sheets and then do some manual/vba stuff to tidy them up and consolidate them; or
2. Write the vba to do the export to a spreadsheet yourself.
Neither is particularly palatable, is it? It's been a while since I wrote some vba timescaledata code but, iirc, it was a bit of a slog to get it working.
You could try this: either use VBA or (if you are using P2000+) create a equation on a Text field to concatenate the Project field and the Task Name field and store the result in the Text field; then do a copy and paste from the text field to the Task Name field. Not particularly elegant, I know, and since I don't have P2000 on this machine I can't confirm that it will work.