I wouldn't do it. Use virtual PCs instead. It might take a little setup, but you'll thank yourself in the long run. In my experience, you can't really get them cleanly installed side by side and unless they are standalone installs you can't really verify that it is 100% true-to-browser rendering.
Update: Looks like one of the better ways to accomplish this (if running Windows 7) is using Windows XP mode to set up multiple virtual machines: Testing Multiple Versions of IE on one PC at the IEBlog.
Update 2: (11/2014) There are new solutions since this was last updated. Microsoft now provides VMs for any environment to test multiple versions of IE: Modern.IE
It is impossible to change the format
We have to differentiate between the over the wire format and the browser's presentation format.
Wire format
The HTML5 date input specification refers to the RFC 3339 specification, which specifies a full-date format equal to: yyyy-mm-dd. See section 5.6 of the RFC 3339 specification for more details.
This format is used by the value HTML attribute and DOM property and is the one used when doing an ordinary form submission.
Presentation format
Browsers are unrestricted in how they present a date input. At the time of writing Chrome, Edge, Firefox, and Opera have date support (see here). They all display a date picker and format the text in the input field.
Desktop devices
For Chrome, Firefox, and Opera, the formatting of the input field's text is based on the browser's language setting. For Edge, it is based on the Windows language setting. Sadly, all web browsers ignore the date formatting configured in the operating system. To me this is very strange behaviour, and something to consider when using this input type. For example, Dutch users that have their operating system or browser language set to en-us will be shown 01/30/2019 instead of the format they are accustomed to: 30-01-2019.
Internet Explorer 9, 10, and 11 display a text input field with the wire format.
Mobile devices
Specifically for Chrome on Android, the formatting is based on the Android display language. I suspect that the same is true for other browsers, though I've not been able to verify this.
Best Answer
Since HTML5 hasn't been completely standardized yet, not all browsers do support these input types. The intended behaviour is to fall back to
<input type="text">.↪ See here for more information on browser support for HTML5 input types.
You can test for support using the Modernizr library or with some custom JavaScript. If you detect a certain HTML5 feature isn't available, you can fall back to JavaScript-based alternatives.