Extensions blocked in High Sierra

Hard drives do not need third-party software to work correctly.

However, if you want to use their crappy bundled tools such as the backup software, then you would need to install their software. I imagine the extension is the handle the "one touch backup" or whatever buttons or displays are on the Seagate drive enclosure.

If you don't use any of those "features," then uninstall the bundled software so it doesn't prevent you from using your hard drive normally.

Additionally, since drive manufacturers think everybody is stupid, they format their drives for Windows and provide crappy software that allows a Mac to write to a Windows formatted drive. If you leave the drive formatted for use on a Windows PC, then you'll need to install their software which will break every time you update your Mac's OS. If you have no intention of using the drive on a Windows PC, then use Disk Utility to erase the drive and format it for use on a Mac. Then, uninstall all of the bundled Seagate software.

If you wish to use the external hard drive as an external hard drive, then you don't need any of Seagate's software. If you want to use it like a toaster, or some other appliance seagate has dreamed up, then you will need their software installed.

High Sierra has greatly changed how third party extensions are used, or allowed. This is all in the rather obvious move with each subsequent major release of the OS to lock users out of the System folder entirely, making the macOS much more secure against malware, as iOS is.

There's a full, and excellent explanation of the incremental steps Apple has introduced to securing the OS here. The article focuses on kernel extensions. The main point being badly written kernel extensions can take down the OS. Not like a badly written app that will (usually) only cause that app to crash.

The point of all of this being that it may be an unsigned kernel extension, which System Integrity Protection will not allow in High Sierra. But, as Barney-15E noted, you don't need to install any of Seagate's junk to use a drive. The OS is perfectly capable of reading/writing a drive all on its own. When you start installing completely unnecessary software that attempts to handle what the OS is already doing, you're only adding a layer of complexity, increasing the chances something will go wrong during a disk operation.

If you used Seagate's uninstaller, you shouldn't have to do anything else. Particularly if it told you a restart was required. That's your clue it removed the kernel extension. The restart is so the OS can update the kernelcache.

An older kernel extension that has not been codesigned will not be allowed to run. Period. It may for a day or so, but SIP will disable it. Codesigned kernel extensions still need to be allowed by the user to run. When you open the System Preferences and click on the Security icon, you'll see a message under the General tab that software by (developer name) wants to load. If you know what it is, click the Allow button and restart. After that, the OS should deem that kernel extension is okay to use.

I've just recently been in an email conversation with Thomas Reed (the author of AdwareMedic, now MalwareBytes for Mac). He noted that even with a codesigned kernel extension, a developer now has to get express permission from Apple to allow it to be used. For that, you need to apply directly to Apple and explain why your app requires a kernel extension, what it does, and what you need it for.