Have you tried setting Unicode Alias to Yes in Global Configuration?
Here, if I do so, I get:
http://localhost:8888/trunkgitnew/component/tags/tag/21-альбом.html
for

tag was created as:

No, I am always avoid using Unicode alias to keep URLs in Latin.
All com_content aliases automatically transliterates into Latin symbols when I'm assigning Russian localization. I.e. Альбом gets albom alias. But problem not with saving or copying, the problem is in typing a word in Cyrillic.

As you see, and taking the alias out of the scope, I had no issue here.

Did you type each letter or just copy / pasted this word? The problem appears when I am trying to input each symbol as I shown it at animated screenshots. Sometimes it's possible to input a word, in some case is not. Kind of floating bug.

Sorry, I see now that it is not in the manager that you have issues, but in the dropbox.

I'll ask other users from local community to check out this issue to be sure it's not problem of my computer, but common problem. And come back with info. Thank you.

I hereby certify that the problem occurs, when values are entered in the national, non-English keyboard layout, only with manual set of values from the keyboard (not paste from clipboard) and then acquired by Enter. (translated into translate.google.com)

Tested by changing my keyboard to Cyrillic and I confirm we do have an issue when creating a new tag in the "tag" type field.
Here, after typing 3 characters, the 4th one does display for a short time then does not display fully.
Going on typing does work although we do not see the glyph.
When finished typing, I had to use the top arrow on the keyboard to move back up to the line where I entered the new tag. Then the full text entered displays and hitting enter does work fine.
I guess the issue comes from the ajaxfield method.

Thanks @phproberto for looking at this.

reason of the problem with word "Альбом" is that char б have same keycode as , (188) in english layout,
the tagajax field check this keycode for "add new tag"

@Fedik
I had the same issue when not using б

@infograf768 seems here a couple issues, first:

after typing 3 characters, the 4th one does display for a short time then does not display fully

and second with б, , keycode

for first issue could be need to hack Chosen.js, in some reason it do not see the value and so reset the width

and about, keycode conflict, not sure how to fix it in b/c way, more simple it just remove check for code 188

After over a year and no suggestions how to resolve this I am closing this,It can always be re-opened if smeone has an idea how to resolve this