This question is very similar to this one, but not a duplicate…in my opinion anyway.
I am not so much interested in finding out which fonts are used, I just want to know if it's possible to package them, much like you can in InDesign and Illustrator.
This seems like yet another oversight and incontinuity between Adobe programs
I know this was asked over a year ago, and in that time I still don't see a way to easily collect the fonts used in a PSD through Photoshop.
However I did think of a workaround, and it worked for me.
Open your PSD file in Illustrator and select Convert Layers to Objects, which will keep text editable (when possible).
Then, package the PSD file through Illustrator.
That should give you all of the fonts.