Select matching element/rename HTML tag in Visual Studio Code

V1.41 is adding this functionality, see https://github.com/microsoft/vscode-docs/blob/vnext/release-notes/v1_41.md#html-rename-tags

HTML rename tags

You can now use F2 to rename the opening/closing tag pairs in HTML.

F2 when the cursor is over one of the tags and you will get a little input box with the cursor to input the new tag name and the start/end tags will be replaced with whatever you type upon .

See https://github.com/microsoft/vscode-docs/blob/vnext/release-notes/v1_42.md#html-mirror-cursor-off-by-default

Also of interest might be the "mirror tags" functionality just added in v1.41 as well (https://github.com/microsoft/vscode-docs/blob/vnext/release-notes/v1_41.md#html-mirror-cursor):

Clicking inside a tag will create another cursor in the matching start or end tag.

VS Code now adds a "mirror cursor" when you are editing HTML tags. This behavior is controlled by the setting html.mirrorCursorOnMatchingTag, which is on by default.

---------- v1.42 is changing the default status of the mirror cursor:

HTML Mirror Cursor off by default

We have made Mirror Cursor an opt-in feature. In the upcoming iteration, we'll continue to improve its implementation to make this feature more easily understandable and available to more languages. You can still use this feature by turning on html.mirrorCursorOnMatchingTag.

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Thanks to JerryGoyal's answer below (I have upvoted it) - I have continued with this ongoing answer.

Because I have been tracking this for over half a year now, v1.44 has renamed this once again. From Synced Regions:

Synced Regions

We have improved the mirror cursor feature introduced last November with a new implementation called Synced Regions. Currently this feature is available for HTML and you can try it out by one of the following ways:

Running the command On Type Rename Symbol on an HTML tag (bound to Ctrl+Shift+F2 by default).

Turning on the editor.renameOnType setting and move the cursor to an HTML tag.

The red regions are Synced Regions. As their name suggests, any change in one region will be synced to other regions. You can exit this mode by either moving your cursor out of the regions or pressing ESC. Additionally, typing or pasting any content leading with a whitespace in any region exits this mode.

We look forward to providing an API that could make this rename-on-type experience available to other languages such as JSX, XML, or even local variables in TypeScript.

As that last part notes, it works in html out of the box but other languages need to implement themselves. As of June 2020 by my testing it still does not work in jsx files on embedded html tags.

You must enable this in your settings, it is off by default.

You can do this without an extension using Emmet Update Tag

1. Place your cursor in the opening tag
2. Press CTRL+SHIFT+P to open the command palette
3. Search for "Emmet: Update Tag" by typing something such as "em up t", and/or find it in the list
4. Press enter to select "Emmet: Update Tag"
5. Enter the new tag
6. Press enter

The opening and closing tag are updated to the new one.

Update Mar 2021:

No need for extension, this is now cooked into VSCode.

"editor.linkedEditing": true

Read more here: https://code.visualstudio.com/Docs/languages/html#_auto-update-tags

Do you want to rename the paired tags? If yes, there is a much easier way: you just need to install the Auto Rename Tag extension. When you rename one HTML tag, it will automatically rename the paired HTML tag.