When you say "this" 5x5 program I don't know which you mean, as there have been very many such things.
On questions as to whether weight stays the same between sets and so forth, rather than take the approach of "It has to be done this way," "No this is the way to do it," "What are you fools talking about, it should be like this," etc, I would rather simply explain what I see as the relative advantages/disadvantages and reasons for using each variation.
1) Same weight all 5 sets, a weight you can do for all 5 sets, increasing weight with time until missing the last set, then sticking with weight till again getting for all 5 sets.
This method allows greater frequency of training (number of days per week), and more total volume per week. It is a more strength-training approach.
2) Same weight all 5 sets, but reps drop with sets as the routine outcome. You increase weight so long as able to get 5 reps on the first set, or some other principle, rather than waiting on getting 5 reps for all 5 sets.
This does not allow as much total volume of training and may also limit frequency compared to the above. Call it a mixed approach.
3) Going for 5 reps each set but dropping weight each set the minimal amount required to just barely get the 5 reps.
This even moreso will tend to limit total volume of training and, likely, frequency. It may be a completely satisfactory bb'ing approach.
Any of these methods is tilted more towards strength training goals by having more rest between sets; and of them is tilted more towards hypertrophy by having less rest between sets.
The differing ways of approaching those questons, not surprisingly, have somewhat differing outcomes. It isn't that one is "the" way and the others are wrong.