Well I may be able to shed a bit of light on this subject since I have and do own owned poodles and am familar with wool.
In the most technical sense a poodle does not produce wool. That is the exclusive product of sheep. Where the confusion sometime comes in is that of all the dogs persons may have an allergic reaction to poodles tend to irritate the least. Two reasons for this are that the hair of the poodle is tightly interlocked and they tend to drop or shed very little hair at all.
Allergic reactions to dogs are generally caused by one of two or both. Skin dander / hair. Poodles tend to have less of these than any other breed of dog.
There is also some confusion over the terms wool and fleece but that another discussion and for your argument you landlord may win since poodles do not produce wool, they just look like wooly little sheep.LOL
And frankly any landlord who is going to be picky over the hair of the dog you own is not one I would choose to rent from.