Have a look at this excellent blog post by Jake Jones:

https://jakejon.es/blog/adding-a-telephone-link-option-to-the-episerver-link-editor

Thank you very much Tomas for that great resource!

There would be a lot of customization required to achieve this. It's purely the href attribute of an anchor element we have control over here (you can see in my example we're just prepending tel: and doing some validation). This means you'd have to persist your button styling option in the href, and then do some custom implementation when it comes to creating the anchor element to split it back out.

Where exactly do you need this? For example, is it just in rich-text fields? Or on blocks?

Edit: I missed the subject 🤦‍♂️

I feel, although not really supported this is a good use case for a PropertyList (https://world.episerver.com/documentation/developer-guides/CMS/Content/Properties/generic-propertylist/)

I'm not sure what ButtonType is in your solution, but I'm assuming it's an enum and you've got an EnumSelectionFactory (see here: https://world.episerver.com/blogs/Linus-Ekstrom/Dates/2014/5/Enum-properties-for-EPiServer-75/). You then want your button link model to look something like this (note the JsonConverter on the Url property):

public class ButtonLink
{
    [Required]
    [Display(
        Order = 10,
        Name = "Type")]
    [EditorDescriptor(EditorDescriptorType = typeof(EnumEditorDescriptor<ButtonType>))]
    public ButtonType ButtonType { get; set; }

    [Required]
    [Display(
        Order = 20,
        Name = "Link")]
    [JsonConverter(typeof(UrlConverter))]
    public Url Link { get; set; }
}

You also need to create a very simple PropertyList:

[PropertyDefinitionTypePlugIn]
public class ButtonLinkPropertyList : PropertyList<ButtonLink>
{
    protected override ButtonLink ParseItem(string value)
    {
        return JsonConvert.DeserializeObject<ButtonLink>(value);
    }
}

You can then use this wherever you need a collection of links where the button type should be definable:

[EditorDescriptor(EditorDescriptorType = typeof(CollectionEditorDescriptor<ButtonLink>))]
public virtual IList<ButtonLink> ButtonLinks { get; set; }