How can I add Ubuntu as a profile option in Windows Terminal?

You need to change settings

Click "Settings" in right top corner of above image, the file profile.json file will be opened. Then inside the word "profiles" in profile.json file, add below snippet.

        {
            "guid": "{78e390db-1bff-4533-9d7c-20f53d8bafa1}",
            "name": "WSL",
            "colorscheme": "Campbell",
            "historySize": 9001,
            "snapOnInput": true,
            "cursorColor": "#FFFFFF",
            "cursorShape": "bar",
            "commandline": "wsl ~",
            "fontFace": "Consolas",
            "fontSize": 12,
            "acrylicOpacity": 0.75,
            "useAcrylic": true,
            "closeOnExit": false,
            "padding": "0, 0, 0, 0"
        }

Near the word "schemes" in profile.json file, you need to update below:

"schemes": [
        {
            "name": "Campbell",
            "foreground": "#A7B191", 
            "background": "#0C0C0C",
            "colors": [
                "#0C0C0C",
                "#C50F1F",
                "#13A10E",
                "#C19C00",
                "#0037DA",
                "#881798",
                "#3A96DD",
                "#CCCCCC",
                "#767676",
                "#E74856",
                "#16C60C",
                "#F9F1A5",
                "#3B78FF",
                "#B4009E",
                "#61D6D6",
                "#F2F2F2"
            ]
        }

The complete setting file (profile.json) which can be obtained here.

Actually, the WSL here is Ubuntu.

Besides, you can add icons to different types of tabs.

You can add icons for the Ubuntu profile to this folder:

$env:LOCALAPPDATA\Packages\Microsoft.WindowsTerminal_8wekyb3d8bb\RoamingState 

I put some 32x32 PNG in this folder, and then in profile.json I can reference the image resource with the path starting with ms-appdata:// .

The icon is available here: Icons

Then replace the contents of the new profile-withIcons.json file below with the previous settings. profile-withIcons.json

After finishing the contents of the folder is like this:

The final result is:

An even easier option: Go to Settings -> Apps -> Apps and features -> find 'Windows Terminal' in the list -> select 'Advanced options' then choose reset.