One way to do this is to make a custom keyboard layout with Ukelele. It lets you add any icon you want to a keyboard, but also includes in the Resources folder in its download some .bundle files which have a number of the former layouts with their flags.
To install those layouts, put the .bundle file in Home/Library/Keyboard Layouts. To get to Home/Library, do Finder > Go while holding down the Option key.
If you need the Ukrainian layout with flag, try this.
Other apps devoted to solving this problem include Keyboard Switcheroo and Colorful Input Menu Flags.
I have not seen any official Apple notice regarding this change, which occurred May 16, 2022. Of the more than 100 keyboards in MacOS, already more than half used the country/language code type format instead of a country flag. It's possible that recent requests to decouple Russian script input from that country's flag led Apple to decide to move totally to this format.
This change may be a direct result of a companywide effort not to denote languages using country flags on Apple’s part.