This is true.
This is not true. True would be for example "The probability of getting four items for slot 1 is equal to the probability of first drawing an item for slot 1, then slot 2, then slot 3 and then slot 4". Both have probability (1/6)^4 = (approx) 0.077%.
Simulation says getting at least 2 different slots from 4 draws as 99.5%, getting at least 3 different has 83% and getting four different has 27% probability.
The reason is that the result "First slot 1, then slot 2, then 3 and at last 4" is exactly one possible result of all possible results but "getting at least 3 different slots from 4 draws" can be achieved by a large number of different results, which all individually have a small probability, but they add up.