How to access Lightroom Classic on two laptops?

HI, the photo files themselves need to be somehow connected to that other computer. Basically Lightroom makes catalog of photo files you "import" into Lightroom. It does not hold the photo files themselves. SO if the photo files - the actual photos - are on the first computer, Lightroom on the second computer has not "improted" the photos. 

I personally keep all my photos on an external drive. I open LR on one computer, import all the photos from that drive to create the catalog. I can then export that catalog to the other computer and then connect the external drive with the photos on it to the other computer. In this situation, you may need to "link" or tell the second computer Lightroom catalog where the photos are located.  Once that is done you can switch from one computer to the next. 

One method to simplify the process of accessing images on two different computers is to use shared collections of smart previews. There are lots of tutorials on how to do this, so I won't try to go into detail here.

Here you'll find a good articel with many informations about your question

How do I use my Lightroom catalog on multiple computers? | The Lightroom Queen

Lightroom stores all the info in its catalog, so you will need to use the same catalog on both computers. The easiest method is to place your catalog on an external drive, together with the images. Connect the external drive to the computer you want to work on.

Where are the photos stored?