Why do academic journals usually have continuous page numbering?

One of the reasons, if not the only one, is that each year journal issues are typically collected by libraries in bound volumes, which, in this way, have a continuous numbering. At the end of the year, journals would also publish indexes for relatively quick reference.

Or, at least, that is what is used to be.

Below, there is an example of bound volumes (picture source).

Many reasons. It makes the page and volume together a unique identifier for a paper, without having to track the issue. This is very convenient. For example, many journals (used to) publish a master table of contents at the end of the volume, and the unique page numbering made these efforts less confusing.

Sometimes, the issue can just be left out of the reference in a paper. It's less likely to happen these days, but before readily accessible databases and DOI numbers, such things were not uncommon. Unique page numbering made the papers easy to find when you were down in the deep, dark, stacks, making a pile of journals for your trip to the photocopier.

Also, what with similar journal names, special issues, supplements, etc., there was something to be said for the ability to notice that the book you were holding, the one that you were willing to put money on that it held the paper you were interested in, did not include the page number that the paper was supposed to be in. Can't tell you why, but this was a problem that came up with some regularity. I think this saved me plenty of time.