Most plugins are made by third parties, but some are made by core developers or by Automattic, the company behind WordPress.com.
Most of us try to keep our plugins up-to-date as far as compatibility goes, but yes, some plugins can become obsolete. Either from developer inaction or from features eventually being rolled into the core of WordPress itself.
A Personal Example
I've been using WordPress for a very long time. When I first started using the system, there was no way to delete spam comments. You could mark comments as spam, and they'd disappear from the admin screen but stay in the database. After I saw just how large my database had become, I built a quick plugin that would delete old spam comments.
The plugin was named ClearSpam and was hosted in the WordPress.org plugin repository.
After a while, though, the WordPress team came together and added a trash feature to WordPress. This allows you to delete old posts, delete pages, and remove comments (spammy or otherwise). It made my plugin unnecessary - it made my plugin obsolete.
I left it in the repository, though, to help the handful of people who refuse to upgrade from legacy version of WordPress. Yes, there are still live sites out there running WordPress 2.5.
Conveniently, it's easy to see whether or not a plugin is still maintained.
Every plugin lists a "Compatible up to" field that tells you the latest version of WordPress it's been tested under. If you're running WordPress 3.1 and a plugin says it's compatible up to WordPress 2.5.3, it probably hasn't been updated in a while.
Users can also vote on certain configurations - if a plugin doesn't work, you can flag it as broken so everyone else knows. This information is shown in the plugin repository and in the plugin installer when you search for a plugin.
Some plugins come along to replace old ones. In a few cases with really good developers, they do this gracefully so you don't lose any information. Automattic recently released Jetpack, a suite of plugins that replaces some older systems they used to distribute. When you install and activate Jetpack, it deactivates the old systems but keeps your settings and data.
With some systems, though, it's just not possible to keep the information around. So if you switch from one Twitter plugin to another, there's a very good chance you'll lose some information in the transition.
The up side of things is that plugins, at least all the ones hosted in the repository, are open source. So when one developer abandons a system, another interested party can adopt it and keep development moving forward.
Another personal example is an older plugin called WP Publications Archive. I used it on a few client sites years ago, but it hasn't been maintained past WordPress 2.1 and I needed it for a current site. Since there hadn't been an update since 2007, I took it on myself to continue development.
I added WP Publication Archive (I know, subtly different name) to the official repository, maintained the same license, credited the original author, and began improving the system to work with the newer WordPress UI.
Did I Cover Everything?
I presume WordPress plugins are made by third parties.
Most are, yes. But many of those "third parties" are the same people who write WordPress in the first place.
Do plugins have to be kept up to date with WordPress upgrades?
No they don't, but most developers will keep them up-to-date if they can.
Can plugins become obsolete?
If they do become obsolete, say with plugins for Linkedin Share buttons, Twitter Tweet buttons, and Facebook Like buttons, would switching to different plugins cause the count of likes / tweets / shares to be lost?
It really depends on the system ... your mileage may vary :-(
And if plugins becoming obsolete is possible, does it tend to happen?
Developers lose interest, code gets bundled into core, someone gets a new job. It can happen for any reason.
Or are plugins generally kept up to date?
Some are ... particularly the ones written by people who you see often in the forums, in IRC, on this site, or in commit messages for WordPress core.
Or are they open source, so other people would keep them up to date?
Most are open source, but people will only adopt a plugin if there's a lot of interest in keeping it up to date.