You're correct in that Employment Allowance is not eligible for a business where the director is the only employee.
Since you've stated that EEs NIC had been deducted but not the Ers NIC, it sounds like that Xero has been set up to calculate your director client's NIC on the non-cumulative "director as employee" method but that the Employment Allowance flag has not been switched off for the business. Some payroll software are automatically set for the business to claim the Employment Allowance, so you'd have to change the setting for non-eligible businesses.
Under the standard annual earnigs period basis, the NIC due is calculated on a cumulative basis and there will be no employee (EEs) deductions until the annual primary threshold amount of £9,568 is reached, after which EEs NIC will be deducted. There will be no employers contributions until the annual secondary threshold amount £8,840 is reached.
Where the director's payroll record has been set up to deduct NIC the same as an employee (the alternative method), then the employee (EEs) and employer (Ers) NIC will be deducted each month on their pay for that month only. In month 12 (end of tax year) the system will calculate the NIC due for the whole tax year's earnings, then compare it to the NICs paid to date to determine the correct over/uderpayment adjustment to make for the last pay period. With this method, there should be both EEs and Ers NIC deduction this month for the pay amount you've stated.