Because Apple is very far from being a monopoly.

You don't have to be a monopoly to have anti-competitive behavior, like disallowing competitors to compete with you in a market that has sizable market share, even if you own that market.

For example, if they banned any non Apple apps in the Apple App Store, there's still ~60% of the market in other stores (Google Play/Windows Store), but that still isn't allowed and is very anti-competitive.

Try obtaining a different browser on your iThing. It can't be done (unless you build one and sideload it through Xcode, or jailbreak). There are "third-party browsers" out there, but they all use the same engine as Safari, by decree of Apple.

However, the reason why Apple gets away with it while Microsoft doesn't should be obvious if you're at all familiar with Microsoft's history and current market share numbers. (Hint: MS got hit for anticompetitive behavior when they owned ~95% of the PC OS market. Apple owns maybe 15% of the world smartphone market.)

Even Microsoft, who should be really ashamed to do it again, only allows Edge browser engine on Windows 10 Phone/Mobile!

IDC released the mobile market share 2015 statistics the other day:

http://www.idc.com/getdoc.jsp?containerId=prUS40664915

Android: 81%

iOS: 15%

WinMobile: 2%

Are there any iOS browsers available that can be installed via sideloading that do not use the Safari engine?

It looks like Chromium can build for iOS, although without much of a UI. I'm not sure about Firefox, as their official iOS version (which is just a UI for the WebKit engine, of course) makes it hard to search for.

https://wiki.mozilla.org/Mobile/Gecko-iOS links to https://hg.mozilla.org/users/tmielczarek_mozilla.com/gecko-i... which exists, again without much of a UI. The build instructions are in the wiki.