If this is indeed a 16 GB iPhone SE, it has to be the circa 2016 model. AFAIK the least amount of storage available on the newer model is 64 GB.
Apple iPhone SE - Full phone specifications says that this vintage iPhone SE can go to IOS 15. Apple lists the first-generation SE as capable of running IOS 15, although it doesn't specify the 16 GB version.
iOS 15
Although it seems like even the first generation, SE can run IOS 15, here are a few things to think about:
1. The issue you are having is likely due to limited space on the device. I have a similar problem with an iPad stuck on IOS 12. Installing IOS 15 will mean that you will have to back up your iPhone, delete some apps to create enough free space to install the update, then reinstall the deleted apps. Not a difficult task but perhaps a bit time-consuming.
2. Even with IOS 15 installed, some of the flashy new features won't be available, or at the very least, won't work well on that model. Check the list of features and their requirements listed on the Apple page above. It may be that your phone won't support the features you want.