Okay, that didn't work "Disk NO NAME" can not be activated.
It says: Try again with Disk EHBO
That's not a good sign; it sounds like the partition got corrupted. This could have been because the PC did something bad to it, in which case you might be able to recover the data by repairing it, and at least can keep using the stick by reformatting it. Or, it could have been a hardware failure (either by sheer coincidence, or because there's something wrong with the USB port on the PC), in which case it's toast.
Try this: Click the greyed out volume, then click the "Repair" button in the "First Aid" tab (that would be "Herstel schijf" in the "Schijf-EHBO" tab. If you're lucky, it will get repaired, or at least you'll get a message that it couldn't be fixed and it will get temporarily mounted so you can copy files off of it before reformatting. If you're unlucky, the repair will just fail.
If that doesn't work, you could try doing the same thing with a Windows disk utility, and see if it is able to repair the partition. Or, if the data on it isn't that valuable, just reformat it and try again, although I'd be careful in case it's a hardware problem--it might happen again.
Does a USB stick have to be formatted differently to work on Macs and PCs like external hdd do?
Almost all USB sticks come formatted as FAT32 or FAT16 (like the one with trouble here). The MacOS can read and write these formats, as can Windows. So unless you've messed with the formatting, almost every USB stick works out of the box on Windows and the MacOS.
If you've reformatted it as NTFS under windows, or HFS+ on a Mac, then you'll run into the standard issues, but so long as you stick to FAT when reformatting (and a MBR partition scheme), it'll continue to be fine.
