The person at the Apple Store got no idea.
If you really have an iPhone SE, it should work provided you update to iOS 13.
To confirm, tell us what you see in Settings > General > About > Model >
you should see a number starting with A????, if you don't then just tap on that entry once
- Apple Watch Series 5 and Apple Watch Series 3 require an iPhone 6s or later with iOS 13 or later.
iOS 13 is compatible with these devices.
- iPhone 11
- iPhone 11 Pro
- iPhone 11 Pro Max
- iPhone XS
- iPhone XS Max
- iPhone XR
- iPhone X
- iPhone 8
- iPhone 8 Plus
- iPhone 7
- iPhone 7 Plus
- iPhone 6s
- iPhone 6s Plus
- iPhone SE
- iPod touch (7th generation)
Furthermore iPhone 6s and iPhone SE have the same hardware chips : CPU (A9 & M9) & GPU (PowerVR GT7600)