Here, engels claims:
If the technique, as you properly say, is for the most part dependent upon the state of science, then so much the more is science dependent upon the state and needs of technique. If society has a technical need, it serves as a greater spur to the progress of science than do ten universities. The whole of hydrostatics (Torricelli, etc.) was produced by the need of controlling the mountain streams in Italy in the sixteenth and seventeenth centuries. We only acquired some intelligible knowledge about electricity when its technical applicability was discovered. Unfortunately, in Germany, people have been accustomed to write the history of the sciences as if the sciences had fallen from the sky.
Is this a widespread belief among historians of science and a focus of their scholarship? Was history of science done differently in the past (as engels claims in the last sentence)?
Also what books would you recommend that focus on how scientific thought developed in connection to certain economic needs and how the wider social conditions of society affected scientific thought and its development?
There are many "lenses" to look how the history of science has developed, and the "needs of the state" is indeed one of them. It is not an uncommon way to do this kind of history these days. In some cases the impact of the "needs of the state" are indeed very important to understanding why science developed the way it did. There are other ways to look at it, though: influences are not monocausal. So other factors that have influenced the development of science include religion, ideology, and non-state economic forces, to just name a few. And there is some trickiness here in defining what the "state" would mean in this context — there are many levels of governance, and many ways of conceiving of states. But sure, yes, it's an approach, and an important one, though not always the most useful one (understanding Newton's work as a product of state needs, for example, gives some insight, but not as much as understanding Newton's work in the context of natural theology). It can be a useful lens, though it is possible to take it too far.
Within the historiography of science, one sometimes sees two "modes" identified: "internalist" vs. "externalist." An internalist argues that science progresses based on forces internal to science itself, e.g., this idea led to that idea which led to that idea. An externalist argues that external forces (politics, economics, ideology, whatever) drive the scientific changes, e.g., this need of the state drove the development of that idea. There are still people who fall more or less in one camp or the other primarily, though today a good deal of the work in the field is more hybrid than this stark dichotomy would let you believe. In practice it often depends on the specific scientific cases you are looking at, and where you draw the line between "internal" and "external" (one of the primary criticisms is that this line is artificial, since science it itself a social activity). What Engels is advocating for is an "externalist" position, and he decries the "internalist" position "as if the sciences had fallen from the sky." Externalism as an approach, as an aside, was heavily influenced by Marxist-Engels theorization (it is explicitly traceable to Marxist historiography of science, notably that of Boris Hessen, J.D. Bernal, and other Marxist/Communists in the 1930s). So there is a deep connection there.
In terms of how the history of science was done in the past, keep in mind that for Engels that would mean 19th century and earlier. And indeed much of the historiography of science from that time is pretty bad — to the degree you can consider it real "history" at all. The field did not really professionalize until the early 20th century, and most of the people who wrote on it prior to that were scientists (who are good at many things, but writing history is not usually one of them) or people who sought to use the history of science as a weapon of some sort (such as those who wanted to use it to promote science against religion). It was largely not very methodologically sophisticated.
Anyway. There are many, many books one could recommend that talk about how the history of science connected up with different "external" factors such as economics, politics, war, and so forth. For a textbook overview, McClellan and Dorn's Science and Technology in World History is more or less a "reformed" version of the kind of externalism that Engels is advocating — it is a "light Marxist" (in that it takes a lot of Marxist conceptions of history for granted, without getting into any real Marxist theory) interpretation of the history of science and technology, and is not terrible (I often assign it to undergraduates, though I also highlight that this approach has its deficits, but it is a useful antidote to their inherited understanding of the history of science and technology as things entirely separate from the history of society). For a somewhat more methodologically complex take (one which synthesizes the internalist/externalist positions a bit more like a present-day historian of science would do), Bowler and Morus's Making Modern Science is not bad.
Aside from these kinds of textbooks, nearly any book you would choose today from an honest-to-god historian of science (e.g., someone with a PhD in the history of science) is going to be looking at "the wider social conditions of society affected scientific thought and its development" in practically all of their studies — this is an essential part of how scholars do the history of science these days.