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From the documentation

works for any dataset where the ei are numerical lists of the same length.

FinancialData returns {DateObject, Quantity}. One way to convert to numerical

numerical = Normal@data /. {d_, v_} :> {UnixTime@d, QuantityMagnitude@v}

But

ResourceFunction["PersistentHomology"][numerical]

seems to hang at "Reducing Boundary Matrix" 6 / 8