Should I be able to connect an Apple Watch 3 given to me to my new iPhone 14?

Make sure the iphone is on the latest iOS ie 16.1.1 or later.

Make sure the Watch is completely unpaired and erased by the previous owner and with WatchOS 8.7.1

When pairing to your phone do not accept any offer to restore Watch data from backup.

With those three criteria satisfied you have a "sporting chance" of pairing it.

Did you friend to what is outlined in this article before giving you the watch --> What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support