2013 MacBook Pro Trackpad will not click/select.

Does the trackpad have physical movement when you press on it? If the movement is restricted, then my guess is some liquid or crumbs have gotten jammed along the edges or gotten beneath the Trackpad itself. There is not a whole lot you can do if this is the case. I have successfully fit a piece of paper into the opening around the trackpad to clear it out, but this will only work for stuff along the edge.

Once in a while macOS can get into a state where clicks won't register. Try rebooting the laptop to see if it makes a difference.

There is also a possibility the button on the Trackpad has failed, although this is not very common in my experience servicing our organization's thousands of laptops.

You can enable "Tap to Click" in the Trackpad System Preferences so you don't have to rely on the physical button to click.

Another thing to look for is any signs of swollen battery trying to burst out of the case.

¿is the Botton still flat, or has it bulged downward?