For me, the first reading is usually higher (by 20-30, even after a long rest period) and subsequent ones get lower and lower as I calm down from the initial worry about whether I will get a high reading or not.
At the doctor's office / hospital I try to tell them this, but they tend not to care. I think they know it varies a lot, they know about psychological effects, but i) they anyway take it much less seriously than overthinkers like me and some fellow HN-ers would imagine and ii) they may prescribe something and anyway expect it not to do much, and also expect the patient not to follow through with taking it properly etc. Honestly, the whole thing is quite a farce. The painful truth is that generic lifestyle improvements are the biggest bang for your buck, instead of worrying about getting exact and precise blood pressure readings.
The other similar big thing is routine blood tests for deficiencies and cholesterol, iron etc. It can also have huge variance over the year, and often people only do it every one or two years and take it as this extremely solid evidence that you need to take this or that medicine. If we were truly serious about this, we would do several tests, separated by weeks, done with different kit manufacturers at different labs etc.
I think the reason for not doing more thorough testing is implicitly admitting that the results aren't really all that actionable and improved precision doesn't really improve treatments because we have no idea what to really do with the results. There are studies showing correlations/causations of certain interventions on specific markers, and those markers are in turn correlated to some outcomes, but often the "evidence-based medicine" doesn't follow the full chain towards the actual outcome.
The other big reason for not measuring more times is the same that a man with a watch knows the time, but a man with two watches is never quite sure about it. In other words, if you got a test and had a result, you can document this and all is fine.