Yes, there are a number of palettes installed with Inkscape. Click on that tiny triangle-arrow on the far right end of the palette bar, and you will see a menu. In addition, you can install more palettes, even one you created yourself.
I seem to recall a couple of tutorials for creating a custom palette, but I might only be able to find one. Let me know if you'd like a link. Inkscape uses the same palette format as GIMP, which if I'm not mistaken, is GPL. So any GIMP palettes are completely compatible!
Besides that, you can create a different kind of palette, which Inkscape refers to as Swatches. You may have figured this out already. But in the Fill and Stroke dialog, this button
will create a single swatch. If you have already applied a fill or stroke color to an object, you can click that button and it will create a swatch of that color. Or you can change swatch colors later, as well. So this is a way to create a custom palette, sort of on the fly. Swatches palettes can be managed using View menu > Swatches dialog.
As far as I remember, a palette made of swatches is a per-document feature. So some extra steps would be needed to make that palette available for other documents. One way would be to create a custom document template, which includes that swatches palette. There's another way, which I don't know exactly how to do, but others surely will know.
Even though the Inkscape manual is slightly out of date, it stilll will answer many of these basic questions. Help menu > Inkscape manual.
