First, do we know that the date cherry trees blossom is controlled by temperature? This question isn't addressed in the linked post, or in the document it links to with the data. They seem to just assume it must be the case.
However, the fact that the dates show very little change until the late 19th century is reason to doubt that temperature controls the date, since there is other evidence that temperatures (globally, at least) varied significantly over that time.
An alternative is that the date is primarily controlled by amount of sunlight, whose seasonal variation has of course not changed over the last thousand years. Presumably, one could resolve this question by looking at actual temperatures in Kyoto versus flowering date, in recent years.
Second, the variability in the date seems to decrease quite markedly in the late 19th century. That happens to be when the Meiji restoration occurred, accompanied by huge changes in Japanese society, including industrialization. It's also when the source of the dates starts to be mostly newspaper reports.
The initial change in the late 19th century is a narrowing of the spread in dates, with no change in mean. Then the mean date starts to get earlier, declining about linearly starting around 1900. One should note that it is generally accepted that CO2 didn't start to have a substantial effect on climate until about 1950, so a decline starting in 1900 is not obviously attributable to global climate change.
Alternative explanations for the decline include local climate change (unrelated to CO2 levels, but perhaps due to industrialization), changes in reporting, and changes in which cherry trees are being looked at.