In this case, 'custom' means familiarity. If you do something often enough, regularly enough, even if that thing is initially traumatising or shocking or disgusting or concerning (like watching your friend shoot up drugs), you 'get used' to it - you become desensitised.
Imagine if you had a new roommate who, say, cut their toenails every night in front of the telly, you might find it absolutely disgusting, and be certain that no one could convince you otherwise. But fast forward 3 months and you might not even notice it any more. And the housemate would certainly find it a bit strange if you suddenly started yelling at them about it, after 3 months of silence.
In the story of your quote, Watson has watched Holmes shoot up drugs every night for three months, yet it still annoys him as much as the first time. This surprises Watson because in any other situation he would have either been spurred to speak up and protest / make a comment by now, or alternatively, just tolerate it as part of life with Holmes.
Reading the full context of this sentence (or even just the next sentence) immediately clears up any doubt and may be of interest to the other valiant answer-ers.
FULL CONTEXT: (from https://ebooks.adelaide.edu.au/d/doyle/arthur_conan/d75si/chap01.html )
Sherlock Holmes took his bottle from the corner of the mantelpiece,
and his hypodermic syringe from its neat morocco case. With his long,
white, nervous fingers he adjusted the delicate needle and rolled back
his left shirtcuff. For some little time his eyes rested thoughtfully
upon the sinewy forearm and wrist, all dotted and scarred with
innumerable puncture-marks. Finally, he thrust the sharp point home,
pressed down the tiny piston, and sank back into the velvet-lined
armchair with a long sigh of satisfaction.
Three times a day for many months I had witnessed this performance, but custom had not reconciled my mind to it. On the contrary, from
day to day I had become more irritable at the sight, and my
conscience swelled nightly within me at the thought that I had lacked
the courage to protest. Again and again I had registered a vow that I
should deliver my soul upon the subject; but there was that in the
cool, nonchalant air of my companion which made him the last man with
whom one would care to take anything approaching to a liberty. His
great powers, his masterly manner, and the experience which I had had
of his many extraordinary qualities, all made me diffident and
backward in crossing him.