Sounds like an apocryphal anecdote to me.
It seems as likely that the name “God” would have been attributed to the particle due to its hypothesized omnipresence.
No, many fields are OMNIPRESENT in particle/theorhetical physics if I am not mistaken, The concept that particles as we understand them interact with various omnipresent fields is not new or unique to Higgs.
What is unique with HIGGS is that particles interacting with this field is what creates what we perceive as MASS. All fields in PARTICLE PHYSICS to my understanding are treated the same way.. nothing unique about HIGGS, other than its the one that gives you mass.
I don’t think there is any hidden meaning to it, just a nice name that sells well and gets clamped onto.