Go to Lightroom Preferences, External Editing tab. There you can set the colourspace and the bit-depth and the file type that LR will use, when sending an image to Photoshop.
It is also possible to set up a different secondary editor here, or else an alternative setup for editing with PS. You can save a number of "presets" if you want, e.g. you may have one for 8-bit sRGB, another for 16-bit ProPhoto, etc suitably named. These appear as options when you use "Edit In..." as well as the main "Edit in Photoshop" that is associated with the Ctrl+E / Cmd+E key shortcut.
Your image inside Photoshop is probably not really a JPG - and if you want to employ PS layers that you can come back to and modify later, it cannot be a JPG. You'd need either TIFF or PSD.
It is normal (when the PS and LR versions are contemporary with each other) for your image document inside PS to be titled - provisionally, by default - with the name of the source image's filename, and not (as yet) with the filename of this new edit version... until that new filename has been confirmed and put into place, by being saved to disk for the first time.
So if you are editing FROM a JPG, it appears that you have sent a JPG to PS for editing. In fact you have made a new TIFF or PSD, but which has not yet been saved as such.
Thank you Richard appreciate the detailed answer. That solved my problem.