Hi Sage, welcome to the site...
Apologies for the delay in getting back to you.
I would say from the picture the buttercream looks like some sort of swiss buttercream or a meringue type buttercream as when you whip the eggs whites they comes up very shiny and glossy, which is what the cake looks like. If you can try and click on the original picture you have sent and see if it gives you any website advice, that way you may be able to obtain the info from the baker who made it. The cake does look super good so hopefully they haven't used photoshop to enhance the colour.
When making anything like this used concentrated dye for the buttercream, I must admit it is a beautiful colour and would look lovely as your god daughters cake...