Unlock MacBook with Apple Watch

Just to be sure, you have "Mid-2013 or newer Mac"?

I will say, I have 1 Mac where my Apple Watch unlocks my Late 2016 15" Macbook Pro

and I have 1 Mac where Apple Watch does NOT unlock my 2017 13" Macbook Pro

Both running Mojave.

The biggest difference is that I own the 15" and the company I work for owns the 13". The 13" has company specified software such as McAfee, JAMF (enterprise management software; then can take control anytime they want), CrashPlan, and a company app that does other company checks.

Where I'm going is, do you any apps on your system that might be similar in nature?

If you boot into Safe mode, does that change anything?

http://support.apple.com/kb/ht1564

This will not load any 3rd party additions, it will load some more conservative Apple drivers (may cause screen flicker), and it will clear some kernel caches (a cache is saved data in a form that can speed up a program, but is totally redundant to the original source, and thus can be safely cleared).  Booting into Safe mode is just an experiment, but can frequently eliminate any 3rd party interference, or a cached item out-of-sync with the world.  (Verify Safe mode via Applications -> Utilities -> System Information -> Software -> Boot Mode -> Safe vs Normal)

Which Series of Apple Watch do you have and what watchOS are you using?

Make sure you have all the settings correct as listed in this support document, How to unlock your Mac with your Apple Watch - Apple Support

I have a MacBook Air on Mojave 10.14.6, and an Apple Watch 4, with watchOS 5.3.1, and it works fine for me, and the settings look just like the image in the support document.

I am using the same software. Go to Settings/Privacy and Security and tap on General and you will see the option.