Answering your title: Can my laptop and my desktop share an SSD
Yes, you can.
Other answers already covered the potential driver and activation issues.
Those are real. However there is a workaround: use a virtual machine.
Install a working OS and a type-2 hypervisor (e.g. linux+vmware player, windows +vmware, etc) or a type-1 hypervisor (e.g. ESX, Xen). Then as a guest install windows on top of that.
Vmware will provide the same virtual hardware to the guest and there will be no driver issues. It will just work.
There are some downsides though: Now you have to maintain three installations: main windows setup with all your settings, and the OS for both desktop and laptop virtualisation.
And if you use windows to run the virtualisation software then you will need additional licences for those. And you will need the right licence to legally run windows in a VM. (IIRC home editions do not allow this, but e.g. datacentre does).
So yes. Technically possible.
And no: probably not in the way you hoped.
As for the question in the body.
Can I take my SSD and use the laptop with it for few days
without reinstalation and with all the applied software and settings ?
Yes. Though for the title I assumed a nice clean setup intended to share.
This question is now about an already install OS. So while above paragrahs are still valid they might now apply to your specific case.
for few days
This also chances the answer. I now assume a single move to laptop and a move back later. Not something which you are going to do every weekend. (e.g. work weekdays on a desktop and move the SSD during every weekend when going home).
Which means that for this I am just going to answer yes.
The same mentioned issues (might need to reactivate, drivers will change, etc etc) still apply. But it will probably work and if you only want to do this move once then it might be worth the hassle to fix all those drivers when you switch.
Still, do make and test a backup. Preferably a full disk one. And be ready to have the installation break down and no longer work at all. Just because it usually works does not guarantee success for one specific attempt.