Greetings IgorSoika,
iCloud Drive is a great feature and you've come to a fantastic place for help. It sounds as though you use tags to organize files on your Mac and recently started using iCloud Drive. After setting up iCloud Drive you're not seeing the tags sync from the Mac to the Files app on your iPhone as expected.
Documents stored in the Documents and Desktop folders on your Mac should sync to iCloud and include any tags you've created to organize those documents. Since noticing the tags haven't synced to the iPhone have you signed in to the iCloud website on a computer to see if the tags appear there? If not, sign in to the iCloud website to see if the tags appear or not. You’ll also want to create a new file on the Mac, save it to the Desktop or Document folder and add a tag to see if that tag appears in Files app on your iPhone.
You may also find that restarting your devices and updating iOS and macOS helpful in reaching a resolution. To restart your Mac, click the Apple logo towards the top-left followed by Restart in the menu. To restart your iPhone, follow the steps outlined in the article below.
Restart your iPhone
If you need to reference the steps for updating iOS and macOS, those steps can be found in the articles linked below.
Update your iPhone, iPad, or iPod touch
Update macOS on Mac
Let us know how that goes!