Unable to change font size

If I use these arrows to change the font size, it rounds them. So instead of 119,0001 > 119, 9998 I get 120.

Does restarting your machine or resetting the Preferences help?

https://helpx.adobe.com/illustrator/using/setting-preferences.html

I agree with you, the increment is not behaving the right way, Does that happen in every document ?  copy the elements into a new docuemnt and check if you get the same result.

to get up to speed you can also use shortcut to increase/ decrease the font size:

ctrl + shift + >  

ctrl + shift + <

increment / decrement by 10:

ctrl + shift + alt + <

ctrl + shift + alt + >

( * On mac use cmd instead of ctrl )

When your cursor is in the Font Size box, If you hold down the Shift key while hitting the up arrow key, you will get larger increases; down arrow key for larger decreases in font size.

As Brad said, it is happening. I can also see the change. 

But it is changing... It shows in your video. Although you are increasing essentailly 120 point type by 1 point increements, so that's not a big noticebale change, but it is definitely happening.