Can I update my iPhone 6 to iOS 13?

You cannot pick and choose an iOS version. What you get is Settings > General > Software update > [All options are available here]

You need to correctly Identify your iPhone model - Apple Support

These are the iOS options based on iPhone Model -->

• iOS 16.3.1 —>
  • Chipset A11 - 8 (2GB RAM) / 8 Plus (3GB RAM) / X (3GB RAM)
  • Chipset A12 - XR (3GB RAM) / XS (4GB RAM)/ XS Max (4GB RAM)
  • Chipset A13 - SE (2020) (3GB RAM) / 11 (4GB RAM)/ 11 Pro (4GB RAM)/11 Pro Max (4GB RAM)
  • Chipset A14 - 12 Mini (4GB RAM) / 12 (4GB RAM) / 12 Pro (6GB RAM) / 12 Pro Max (6GB RAM)
  • Chipset A15 - SE (2022) (4GB RAM) / 13 Mini (4GB RAM) / 13 (4GB RAM) / 13 Pro (6GB RAM) / 13 Pro Max (6GB RAM)
  • Chipset A16 - 14 (6GB RAM)/ 14 Plus (6GB RAM) / 14 Pro (6GB RAM) / 14 Pro Max (6GB RAM)
• iOS 15.7.3 —>
  • Chipset A9 - 6S (2GB RAM) / 6S Plus (2GB RAM) / SE (1st Gen) (2GB RAM)
  • Chipset A10 - 7 (2GB RAM) / 7 Plus (3GB RAM)
• iOS 12.5.7 —>
  • Chipset A7 - iPhone 5S (1GB RAM)
  • Chipset A8 - iPhone 6/6 Plus (1GB RAM)
• iOS 10.3.4 —>
  • Chipset A6 - iPhone 5 (1GB RAM)
• iOS 10.3.3 —>
  • Chipset A6 - iPhone 5C (1GB RAM)
• iOS 9.3.6 —> iPhone 4S (512MB RAM) - Chipset A5
• iOS 7.1.2 —> iPhone 4 (512MB RAM)- Chipset A4

Know how to...

• Update Over-the-Air (OTA) —> Update your iPhone or iPad - Apple Support (IN)
• Update using iTunes/Finder —> Update your iPhone, iPad, or iPod touch - Apple Support (IN)