While Jon did have Ned as a fatherly figure. He still had a somewhat difficult childhood. While Robb and Sansa probably saw him as a brother as they were pretty close in age with Jon being the oldest IIRC. It seems it was made quite apparent to him that he was a bastard with no place. He was part of the ruling family but really not. Catelyn treated him pretty badly it seems as she admits she never loved him.
It was probably drilled into his head from an early age that he was going to take the black to ward off any potential succession issues.
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