You swapped the main ingredient from a fruit to a liqueur. How is it related to this cocktail ??

Were the juice and coconut milk cold?
If not, it takes a lot of ice to cool it down.

That's what just happened when I made one - alcoohol-free - for my kids!

If you have some ginger handy, you can crush some with lemon juice, add a bit of sugar and top up with soda water, and w-bam you have an ersatz of ginger beer to your taste.
I can't give you exact measurements, but I'd say a thick slice of ginger is enough for one serving of ginger ale-ish.

I had some cheap ginger ale (Schweppes Canada Dry) and tried it; it was too sweet and way too bland.
In the base of the shaker, I muddled a small slice of ginger root, with about hald a shot of lemon juice, and added it to the mix: I find it more balanced, and the ginger makes up for the lack of expressiveness of the cheap ginger ale.