Student or Learner
What is the grammatical reason behind saying "equal sign" instead of
"the equal sign"? Are they different in any way?
Other examples I found include "I am in study mode" instead of "I am in a study mode" or "I am in the study mode" although the last example would have to be specified.
Is there any more examples of these kinds of usages? and and how do they sound differ to a native speaker?
You would need to provide context, but one common usage would be an equal sign, when you are referring to a instance of the symbol. I wonder if you're thinking of the rule of using the definite article where there is only one of something; if so, then remember that this is not an absolute rule, and with the equal sign, there is only one, but if you're writing a formula, you're often using an instance of it. Article usage is complex and flexible, and does not follow rigid rules that can be applied in every case.