Every now and then I read about people who wonder whether zero is a number. It never occurred to me to question this, so I checked the Wikipedia page which, when talking about the Rules of Brahmagupta explains
In saying zero divided by zero is zero, Brahmagupta differs from the modern position. Mathematicians normally do not assign a value to this, whereas computers and calculators sometimes assign NaN, which means "not a number."
I did consider whether this difference in position may be the reason why some people state that "Zero is not a number".
A few distinctions:
In your first link, the discussion is regarding whether $0$ is a natural number, not that it is/isn't a number.
Your second link is predicated upon fairly weak, insubstantial first-year undergraduate arguments that can be dispelled with a more rigorous construction of numbers. I also question any blog that concludes a logical argument with "what's your opinion?" (I'd also question the veracity of a mathematical argument coming from a theological blog -- in the same way that I would question the veracity of a monetary policy argument coming from a sports blog).
Finally, the $0/0 = 0$ argument is considered non-standard. That most mathematicians define $0/0 = \text{NaN}$ is not the same as $0 = \text{NaN}$ because these are competing definitions.
So, in short, people often claim "zero is not a number" because they lack the background to understand the formal, rigorous definitions of the number system.