On P356, q is defined as "multiplier" with the unit of $ and the multipliers of the future contracts in elsewhere in this reading are always with the of $.
Since I think that the futures prices are with the units of $ (as f on P356 and elsewhere in this reading) and this shall be a common sense, then why "multiplier" can be with the unit of $ ?
This is from last year's discussion. So I don't think CFAI will correct it any time soon. It may have been there since the index futures were invented. For example, S& 500 Futures' Contract Size: $250 x S& 500 futures price.
For index futures, the future price is just a level and has no unit. The multiplier in $ is usually fine, but it does cause problem when we calculate the number of [synthetic] units of stock.