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In general, tax liability on the settlement funds exists independently of where the settlement funds are kept, so there is no reason I know it must be kept in his account.
Note that I am not a tax specialist, and obscure tax regulations abound: ask him to explain the tax liability he's referring to and why it prevents using an independent escrow. There's no reason for not explaining, and his reluctance to do so seems odd.
Naturally, lawyers will have tax liability on any portion of the settlement that was agreed to be used to pay fees, just as the ultimate recipient of the settlement funds may have tax liability.
Of course, the funds themselves will generate tax liability on the interest raised in whatever account they reside.