Yes, getting only 16 colors is a classic symptom of the video driver not being properly installed. This used to be a big problem on Windows 9x; the default driver only did VGA (640×480, 4-bit color), and most contemporary systems did not have drivers bundled with the OS. I was working as a technician in a repair shop around this time, and remember being massively relieved when Windows 2000/XP became popular enough to replace 9x, as they had virtually all necessary drivers bundled with the OS.
The normal way to solve this problem is by going to the manufacturer's support website and downloading the appropriate driver. Here is Sony's driver download page for that notebook. The pre-selected default is Windows 98 SE, as that's the OS the computer shipped with. That's a good sign, because it suggests that there will be a Windows 9x-compatible driver.
Unfortunately (and oddly), there's no video driver listed on that page, either for 98 SE or Me. I don't know why that would be omitted. Lots of other drivers are there (you'll probably need those, too), but no video.
There is a video driver available for Windows 2000, so you might want to try that one—you might get lucky and find a Windows 98 driver included in the package. Maximize your chances of success by not using the install package bundled with the driver. Instead, unzip it to a temporary directory, go into Device Manager, update the driver for the unidentified video device, and point it to the temporary directory into which you expanded the driver data.
If that doesn't work (and improbably won't), then you'll have to go hunting for a driver out there on the web. This can be a lot of work, and requires a lot of patience. It can also be somewhat dangerous, as you're downloading binary executable files from unknown sources. Caveat emptor. This website has a long list of drivers related to that one, including several for the "256XL", listed as supporting Windows 98. Before giving up, it would definitely be worth trying a couple of those.