Any English teacher using the expresssion should be sacked.
At best, and it's pushing it, "Everybody is special" might convey the idea, rather poorly expressed, that everyone is an individual.
It's much more likely to represent fuzzy thinking, or the avoidance of thinking at all.
Not everything can be special. It's a linguistic nonsense:
"What's your speciality?" "I specialise in everything"
It's on a par with "make everything a priority".
It's camouflage, trying to protect the speaker from having to acknowledge different abilities and priorities. Which exist.
In any group setting (from family to classroom to planet!) there are the needs of the group, and the needs of the individual. There will therefore be a degree of inherent conflict. Negotiating that is a basic issue of civilisation.
Denying it is to engage in fantasy.