Celtic is a Latin word (Celtae) from Greek (Κελτοί—Keltoi). In classical Latin, all Cs are hard. However, later forms of Latin (e.g. ecclesiastical Latin) softened the C. So a name like Caesar pronounced in classical Latin is Kaesar while in ecclesiastical Latin it is pronounced Saesar.
For whatever reasons, the basketball team decided to use the ecclesiastical pronunciation. Modern English typically does, and perhaps more so back in the 1940s when the team was founded. However, people instead chose to use the classical pronunciation when referring to the linguistic group of people. Perhaps they wanted that name to reflect what the Romans that named them. If there is an exact socio-cultural reason aside form arbitrary choice, I am not familiar with it. Perhaps an etymologist or linguist can chime in and some more history behind this.
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