Sorry for not coming to closure on this. The problem is not difficult, and I think we're missing each other re: communications. There were some questions left unanswered, so I assumed acknowledgement by silence.
Here's what I think is going on. Your files are imported into MC on the desktop system. MC's database is correct, showing you albums, etc. But the file tags themselves do not contain the same (good) meta-data. So you copy the physical files over to another drive, and then use a new MC library and import. MC gets its preliminary information by reading the tags in the files. If these are incorrect, the new MC's library will reflect that.
So, what you need to do is focus on one single album that exhibits the problem. Just one. For that album, which groups correctly on the Desktop system, select all its tracks and right-click Library Tools > Update tags (from library). Now, recopy just that single album using Rename, Move & Copy (or you can just manually copy the actual tracks in Windows Explorer). Bring the external drive to the laptop, use your MC there, and go select those same tracks, and select Library Tools > Update library (from tags). Hit F5 to refresh the view. Let's see if this solves the problem.
Don't worry, there are plenty of folks who can help.
Finally, if we don't make headway with this, remote help may be an option.