Can my laptop and my desktop share an SSD
Yes, you can.
Is it a good idea? No, and it may not work either.
When you take out the SSD from your main desktop and plug it into your laptop, the SSD may be connected to a different physical port internally. The boot manager may therefore not be able to boot from the SSD, giving a BSoD in return. It can be fixed, and some PC's are smart enough to do so for you if you desired. If your laptop is able to correct the problem, but your main PC is not, then your SSD will not work on your main PC anymore.
Furthermore, assuming things work correctly, Windows will install drivers for the laptop, polluting your system, making it slower.
Not to mention that if your drive is encrypted, it will most likely not even work at all.
So my advice is: don't do it. It's only for a few days, and the mess you can get from it is definitely not worth it. You may end up not having a laptop with you at all.
In addition to the previously mentioned driver issues, the Windows licensing process will usually recognize that it is on a new computer and will deactivate itself. If you have a retail license, you can deactivate it on the old computer and activate it on the new one. If you have an OEM license or the free upgrade license (most people), then you would have to buy a retail license.
This has a good expanded explanation, covering both the driver issues and the licensing issues: https://www.howtogeek.com/239815/why-cant-you-move-a-windows-installation-to-another-computer/
I'd be doing a full backup to an external drive before doing that. You'll add a ton of drivers to your system that you don't need, which can have bizarre consequences later. (Usually performance issues.)
For years I have used a program called EaseUS Todo Backup to migrate OS installs to new hardware - sometimes you need to for businesses that can neither reinstall nor reactivate legacy software, because the company is gone/dead. It works on pretty much any version of Windows. It removes the most problematic drivers from the image, allowing them to redetect on first boot, which lets you (for example) backup an Intel/IDE system and restore to a modern AMD/AHCI system, and have all the new drivers autodetect and configure properly... vice versa also works. That said, there are usually performance issues or other quirks. For a business, they wouldn't care. For a gamer, you certainly would.
That said, Win10 now has some of the same functionality built in. It fixes IDE/AHCI drivers when you go into safemode, for example, so just locate the Advanced Startup section in the recovery console and use that to fire into safemode once and it'll fix that one for you.
https://www.howtogeek.com/126016/three-ways-to-access-the-windows-8-boot-options-menu/
They also added a fix for a common BCD problem to the Restart button, so when it says you can "Click Restart to try again, yada yada", it's actually applying fixes, and sometimes that saves you a trip into the recovery console to use tools like bootrec or bcdboot. Click that Restart button at least once for all BCD/AHCI/etc problems.
Now, all of that said - I personally wouldn't do it on my main gaming PC without a backup, because then I might have to reinstall Windows and all my games/software to fix a performance anomaly or other glitch. That's a huge hassle. Far more hassle than a new $50 SSD.