The customer is not necessarily right.
He is correct in that just because Poker can be played with a deck of another size, does not mean the cards of that size are "poker size" cards. However, he is incorrect in that there is exactly one set of dimensions for a playing card that are inviolably and universally standardized as "Poker size". Just in the question, Roaring Fish's answer and aramis's comment, we've gotten three unequal dimensions.
Wikipedia states that the standard for a "poker size" card is 2.5"x3.5", which when converted to metric is 63.5x88.9mm. That same paragraph states that this is a B8 card; however, the ISO B8 paper size is exactly 62x88mm. So, right there, you have two different sizes based on who's doing the measuring; a U.S.-based printing house, or a European or Asian one. Conversion of the U.S. measurements to an exact millimeter value produces more very close but not quite equal measurements: the width could be 63 or 64mm, while height could be 88 or 89mm. In any case, the U.S. poker size works out to a larger card than a true B8.
Now, you might say that these differences of a millimeter or two either way are insignificant, and thus the customer is right; they're the same size. It is indeed a small difference for the player or the game designer, but printers live and die on smaller measurements than that. In this case, the ISO standards for paper and card stock measurements are used to sell the raw materials in Europe, so a printer might be starting with a B0 sheet of card stock (1m x 1.414m) to print the cards for your game. Each larger number in the size series represents a halving of the longer dimension of the next larger size, to create a sheet half the size of the larger. Producing true B8 cards from a B0 sheet of card stock is thus a relatively simple matter of dividing this sheet by 16 (2^4) in each dimension (rounding down, as some material is invariably lost to the cutting process), giving the printer 256 B8 cards per B0 sheet.
However, if the publisher were to insist, as the customer here is inferring, that poker-size cards are exactly 2.5x3.5" and no other size will do, now the sheet can only fit 15 cards on a side or 225 total, because the extra 1mm per card makes the B0 sheet a full quarter card too narrow, and 10mm too short, to have the extra 31 cards on the same sheet. That increases waste per B0 sheet from a mere 48 square millimeters per B0 sheet (inherent in the printing/cutting process) to 3750 square millimeters of paper you're still paying for that's now going to the landfill. More importantly, it means that a print run of a game with more than 225 cards will require a second printing line to be tooled, adjusted, proofed and manned to print the sheets with the extra cards, and that dramatically increases print costs, as well as the handling costs of splitting, sorting and packaging the extra sheet's worth of cards into the game boxes (and the intangible costs of the game being thought of as a screwup by customers should the packers get it wrong). These changes will definitely lead to a printer charging the publisher a higher cost per copy, and that can influence the decision of which printer to use, increasing the pricepoint of the game. If the size of the game box has to change in order to accomodate the just-slightly-larger cards (uncommon with board games, but card game boxes are a pretty tight fit to the decks inside), that can throw off the pallet stacking, so fewer boxes can fit on a standard pallet, meaning you're now spending money to ship the air between the game boxes instead of the game itself. All for 1mm a side in card dimensions.
So, Mr. Customer, there are at least 6 different permutations of exact measurement that are feasible for a "poker size" card, and the exact size of the playing cards you want can in fact make or break the financial success of your game. Whether it will or not, we can't know until you give us exact specifications, at which time we can design the printing plates of the game and give you a number, and possible ways to reduce inefficiency and save you money if you're willing to alter the game design.