I wouldn't say I'm more widely read (any more, at least), especially when concerning Shakespeare, but I wonder if they were referring to his poem "The Phoenix and the Turtle"?
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Shakespeare died in 1616 according to Internet, and may not have been alive by the time the legend of Thunderbird reached Britain?
Ime, the concepts of the Phoenix & the Thunderbird often seem to be conflated, even though they're quite dissimilar, and come from pretty significantly different cultures - but I don't know if that's the situation you heard?
Unrelated, but it makes me slightly sad that the Thunderbird bass isn't in the "In Modern Usage" section of the Thunderbird Wikipedia page