CAT5, CAT5e, and CAT6 are ratings that apply to the cable, the connectors, the termination, routing, etc. You want to rate CAT6, you do the whole shebang. This is not speaker wire, it is high-performance networking cable. As soon as you put CAT5e connectors on, it is a CAT5e system.
According to engineers: ""Although Category 6 and Category 5e connectors may look alike, Category 6 connectors have much better transmission performance. For example, at 100 MHz, NEXT of a Category 5e connector is 43 decibels (d, while NEXT of a Category 6 connector is 54 dB. This means that a Category 6 connector couples about 1/12 of the power that a Category 5e connector couples from one pair to another pair. Conversely, one can say that a Category 6 connector is 12 times less "noisy" compared to a Category 5e connector. "
Will this affect us? I don't know. Our systems are going to demand higher and higher bandwidth, and we will stress the cables, but who knows whether the marginal benefit is there. As reasonable people, we might make the decision it doesn't matter.
But I don't like that we are giving advice that putting in CAT6 connectors on CAT5e wire is fine, or that it is just a matter of thicker cables or staggered punchdowns. Certification does matter, and when it says CAT6, it should be at least a reasonable attempt at CAT6.
We are no longer in a world of analog and amperes, and we should resist the urge to conflate the new digital world with the old analog one.