IntelliJ Shift Shift Shortcut in Visual Studio (Global Search)

Not exactly the same (doesn't show the last used/opened tabs), but works to jump to a file (might be VS 2013+ only):

CTRL+, then type your search terms, you can use space for different parts of the file name:
con na ex => ControlNameExplorer

I find it easier to use when disabling the Preview Tab (click on the down arrow at the right of the search box to change settings).

More details here.

EDIT
This shortcut is set when installing ReSharper, but you can manually set it to the command Edit.NavigateTo (Tools > Options > Environment > Keyboard).

EDIT for VS2017
The command is now named Edit.GoToAll. Thanks to Matt for pointing it out.

No need for ReSharper since Visual Studio 17 has already implemented a similar feature:

CTRL + T

It's became possible after the recently VS Code release(1.54.0 or above), you can update keybindings.json with following code:

[
{
    "key": "shift shift",
    "command": "workbench.action.quickOpen"
},
{
    "key": "alt alt",
    "command": "workbench.action.quickOpen"
},
{
    "key": "ctrl ctrl",
    "command": "workbench.action.showCommands"
}
]

Source: https://github.com/microsoft/vscode/issues/5280#issuecomment-767869212

Or install VS Code extension: shift shift