Well, I have up to date everything as I stated in my first post.

@gabriel_nemeth  actually what you said in your first post was "the Dell drivers are up to date".  My suggestions were firmware updates, not drivers.  But I would have suggested that anyway because over the course of helping many people here, I've seen several cases where people said, "I've got everything up to date", and it turned out they didn't -- and the update they were missing turned out to fix their issue.  In any case, based on your latest post, if the firmware updates I suggested are already applied and/or don't resolve your issue, then I'm not sure what to suggest.  It doesn't make sense that Windows would only be able to load drivers for a USB external storage device when it's present at boot.

@gabriel_nemeth  that's very strange, particularly the difference based on whether it was connected at boot.  There are some Thunderbolt BIOS settings that come into play there, but they only apply to actual Thunderbolt peripherals, and the Samsung T5 is just USB/USB-C, not Thunderbolt like the Samsung X5.  My only immediate suggestions would be to update the BIOS and then the TB3 controller firmware, even though again the latter shouldn't really come into play when using a non-Thunderbolt device on that port.

Also, I noticed, that when I plug in the SSD after my laptop boots, I got a triangle warning in the device manager:

This device is not working properly because Windows cannot load the drivers required for this device. (Code 31)

Object Name already exists.

Ok, I get it. I thought when Dell Support Assist says everything is up to date then everything is up to date. Honestly, I have no idea how to update bios firmware, because if I search dell  XPS 15 9570 bios firmware update, I find only the bios driver page, version 1.15.0, which is my installed one.

Omg, when I uninstalled that first one from the device manager, it started working. Now it recognizes the t5 finally. But what that first USB was?

@gabriel_nemeth  the latest TB3 firmware update for the XPS 15 9570 as of this writing is here.  And as a possibly useful reference going forward, here is a direct link to the entire Downloads page for that system on support.dell.com.

However, your screenshots finally came through -- initially they're only visible to the person who uploaded them and to Dell moderators until they get "approved" for public viewing -- and it appears that the device having a problem is the actual USB host controller itself, not the hard drive as I figured based on you not specifying anything otherwise.  That's a different story.  I also see that you have a VirtualHere USB 3 controller.  I'm not sure what that is -- possibly something related to USB device redirection/virtual attachment for a virtualization solution or remote application streaming service like Citrix? -- but that reminded me of another similar issue I read about a while ago.  An older Dell dock called the TB16 had a USB controller that required a third-party driver from ASMedia, and eventually it was discovered that if the user also had Citrix Receiver installed -- which installed a virtual USB controller driver to allow local USB devices to be redirected to the remote application -- then it could actually prevent the dock's USB devices from working at all.  Its ASMedia controller would throw an error like that in Device Manager and sometimes even cause system lockups.  The fix in that case was to install a newer version of Citrix Receiver.  So I'm wondering if there might be a similar conflict with your VirtualHere USB driver.  Can you try updating whatever application created that to the latest release?  If that doesn't resolve it, can you try uninstalling it entirely just as a test?

Ok so here is what I did:

• Uninstalled the fist Intel USB 3.1 eXtensible Host Controller
• The XPS finally recognized after that the T5, but after that, I figured out, that the uninstalled device was actually the USB port on the right side. So I installed it back.
• As you mentioned that VirtualHere USB 3 controller can be the issue, so I uninstalled it and it works. I have no idea what that thing was, or which device used it, but I uninstalled it.
• I googled a little bit and find this forum, same case, sadly no solution: https://www.virtualhere.com/content/connecting-storage-devices-allows-detection-not-interaction

So currently I'm fine with that.

Thank you @jphughan for help and support.

@gabriel_nemeth  One other thing to check in addition to the update and tests I suggested above.  If you start your system without the T5 attached, check Device Manager and see if you even have two of those "Intel USB 3.1 eXtensible Host Controller" devices present at all.  If you only have one, then I'm thinking that the XPS 15 might not even start the USB controller that runs its USB-C/TB3 port unless a device is actually attached to it.  And if the underlying cause of this issue is a conflict between the VirtualHere controller and that one, then that might account for the difference in behavior based on whether the T5 was attached at boot or not.  For example, if the T5 is connected, then that second Intel controller might start before the VirtualHere driver loads and therefore allow devices connected to that USB-C/TB3 port to run properly, but if there's nothing connected to that port at boot, then that second Intel controller DOESN'T start before the VirtualHere driver loads, and when it tries to start LATER when you actually connect something, you have this problem.  I suspect you'd also find that this issue isn't limited to the T5, but would instead affect any device you plugged into the USB-C/TB3 port that relied on being able to run USB 3.x data.

Well, I have nothing attached to that first Intel USB 3.1 eXtensible Host Controller. I doesn't even know what that first one is.