Is this a cake or a large cookie?

I think this is a good question because it isn’t clear in the narrative with the recipe. I’ll confess that I am offering an answer to the question even though I haven’t made the recipe. From the photo, and from a comparison to the original recipe for Nestle Tollhouse Cookies (butter, flour, eggs same or nearly same amounts; amounts of sugar and leavening are just a little different) – my guess is that these are more like a thick bar cookie than like a cake. It’s like taking a slightly tweaked cookie recipe and pressing it into a cake pan. I look forward to trying it, maybe a half recipe in a smaller pan for the first time.

Another view.
This reminds me of those skillet chocolate chip cookies in fashion some years ago, which could be broken in pieces to serve a whole dinner party. Even though not round, the pieces were flat or shallow enough to resemble cookies.
This one is like a skyscraper version of that skillet cookie - so deep, we need to slice it in wedges like a cake.

A bakery would probably call it a cake mostly because they could charge a higher price for it. They could charge even more if they call it a gateau or torte. ;-)

The recipe author used both the "cookie" and "cake" tags so maybe she isn't sure herself.

I don't care much for overly sweet desserts myself however I know I am in the minority here in the USA.

Just make it and call it what you want. The most important thing is that your recipients enjoy it, regardless of its classification.

It is difficult to say which camp you would put this in- but I tend to think it would lie closer to the cake side of things. I also am not convinced that the amount of frosting is significant enough to add too much to the sweetness level either- unlike say, a bakery cupcake, or even some of the "fancy" cookies on sale now. I think this would be kin to a whoopie pie, in the sense it is sort of cake, sort of cookie. But you will just have to bake it and see for yourself.