IE11's User-Agent string is substantially changed from older versions of IE, *because* it is a much more compliant browser and newer websites were sending incompatible, or fallback, content. The change makes it look a lot like Chrome, which means most sites will send it their latest content version, which should be mostly compatible in IE11.
Microsoft Dynamics will have to come up with an update that actually detects IE11 as a 'capable' browser before it will work without selecting Compatibility Mode.
The Compatibility Mode button - which is only available on the desktop browser, it's not in the 'Metro' version - tells IE to send an IE7 User-Agent string to the server (nearly - it sends ';Trident/7.0' in the string as a tell-tale that this is really IE11, not 7). The browser then defaults to its IE7 rendering mode, unless the site sends an X-UA-Compatible HTTP header (or META tag) telling it to use a newer mode.
If IE decides that the server you're connecting to is on your Intranet, it will use the Intranet Zone settings. The default for the Intranet Zone is to always pretend to be IE7. This can of course cause problems for applications developed for IE8 and up. The Intranet Zone is, by default, only enabled for domain-joined computers, and the default detection rule is basically 'if the hostname in the URL doesn't contain any dots, it's Intranet'. The Intranet Zone rules are configured on the Security tab of Internet Option - click the Local Intranet icon, then click Sites to set up the rules for what is considered Intranet. To disable compatibility for intranet sites, press Alt+T to get the old Tools menu, and select Compatibility View Settings. Then uncheck "Display intranet sites in Compatibility View". These settings can be set through Group Policy.
Microsoft's Compatibility View Lists also gives them the ability to send a custom User-Agent for specific domains. This is what went wrong with IE11 against Google's websites when it was first release: Google's code didn't work with IE11 originally, so Microsoft added their domains to the compat view list indicating IE10 (but using the Trident/7.0 token rather than Trident/6.0 as IE10 would send). Then, just around the time that IE11 was released, Google fixed their code to work with IE11's real User-Agent, IE10's real User-Agent - but it broke when IE11 sent its pseudo-IE10 string. MS then took Google domains out of the compatibility view list, but it takes a little while for the browser to download a new list.