The description "WebApp" seems to suggest to me that it truly is just a browser, just one that's dedicated to a single site. Legally, there's NOTHING about that that is questionable. Now, the use of the logo in the tile/icon/whatever might be questionable... but if it's the logo used by the FavIcon for the site, I'm not sure there's even anything legally wrong there. If "the mobile experience might suck" that's a first party issue with their own site and "easily" remedied by themselves, so there's little room for complaint, much less legal action. Claiming they had "absolutely nothing to do with" it is simply wrong... it's their web site, they have everything to do with the experience. Effectively, Microsoft created a bunch of one off browsers, and that's it, the content is still solely under the control of the web site owners.
@TheJoe, there is some debate about what web site content can be used for, but it sounds like in this case the debate is meaningless. These apps are just browsers, so the content is being used precisely as it was intended to be used. It may be odd that the browser is a dedicated browser, but it's still just a browser.
Now, to be fair, I'm basing all of what I said on assumptions about what these apps do. I've not installed any of them myself, so I could be entirely wrong about them simply being dedicated browsers. I'm also not a lawyer, so I could be way off in my interpretation of what I know of the law in my own country, but it seems to be common sense to me. Any complaints I've seen raised thus far would apply to any browser, so I don't see how there can be legal reasons for any of this to not be above board... other than violating their own app policies and simple good taste.