Henry Jones senior was, according to backstory material written but not presented in the film, born in the 1860s, and was a Scottish university professor before emigrating to Utah, where Indy was born. He was roughly 75 years old in 1938.
Sean Connery was only 58 at the time of filming (and only 12 years older than
Harrison Ford), hence the beard and general "old man" attire his character wears. Indy impersonating a Scottish lord at Castle Brunwald was a nod to this unspoken backstory.
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