Engineering as a profession is really, really old. If you are asking about engineering education at university level, then that is a newer thing. Speaking about engineering education in the United States, the first curricula copied the French "polytechnique" model of the 1800s. After all, the Ecole Nationale de Ponts et Chaussees was largely considered the first formal engineering school in the world, beyond the more common engineering classes. Of course, Napoleon later founded what became Ecole Polytechnique.

In the US, universities of the 1700s was based on religious education and liberal arts. So US engineers at that time were either self-taught or was educated abroad. This was seen as a weakness in many aspects including military, so much so that George Washington himself strongly encouraged the formation of an engineering school. This led the US Military Academy, aka West Point, to have a strong engineering focus, at its founding very much copying the Ecole Polytechnique curriculum. Several military and civil engineering schools followed, and these had strong focus on workshops and lab work. Some of them are Schools of engineering affiliated with a university (Harvard, Yale, Dartmouth, Michigan) but had their own (lower and less prestigious) admission. So these polytechniques and schools of engineering were either segregated from university or completely apart from any.

The Morrill Act of 1862 established the "land-grant" concept, strongly promoting practical education aka engineering. As a result, many land-grant engineering universities were founded, such as Purdue, Penn State, Iowa State, UIUC, MIT, et al. And further, these universities had schools of engineering well-integrated into the universities themselves. Over time, previously segregated schools of engineering also became integrated into their universities.

Following WW1, engineering education started to take on newer ideas from Europe, namely the strengthening of scientific education to provide a basis for engineering know-how, and reduced emphasis on workshop. All this changed the nature of engineering education in the US, but the larger change was possibly WW2 and the advent of the Cold War, which led to significant government (read: defense) investment in R&D at universities.

• A Brief History of Engineering Education in the United States by LAWRENCE P. GRAYSON, Fellow, IEEE, National Institute of Education; IEEE Transactions on Aerospace and Electronic Systems, Year: 1980, Volume: AES-16, Issue: 3, Pages: 373 - 392, DOI: 10.1109/TAES.1980.308907

So I have done some research into a land grant engineering college in the US (WPI), and that may provide some background into this from a mechanical engineering prospective. A lot of the early professors at WPI were actually just very skilled machinists. It seems, based on the history of the school and the professors at WPI that in the US, a large portion of "engineering" in the early to mid 1800's was done by people who had apprenticed as machine operators and had gone on to improve those machines. The other portion of professors at WPI at that time were people who had been educated in Europe, and were more concerned with the theory, people who insisted that students knew the physics and math of what was going on.

Based on this, engineering seems to have grown from blacksmiths who started making machines and becoming machinists and machinists who eventually became engineers. While simultaneously growing out scientists who started to make machines out of the science they were doing. These types of people started learned from each others expertise, especially after the land grant universities started popping up.

All of my sources should be buried somewhere on the WPI website, and this stuff is probably tainted somewhat by the founders and early teachers of WPI being heavily invested in the idea of "theory and practice," the motto of WPI.

This comment is aimed at engineering as it relates to construction.

The oldest architectural text, De Architectura, was written by Vitruvius who was a Roman military/civil engineer. There wasn't a distinction between master builders, architects, structural engineers, etc. until sometime in the mid to late 19th century depending on where in the world you were.

A good example of someone that really personifies the combination of knowledge areas is Brunelleschi. He built the Florence Duomo (Florence's cathedral). It is, to this day, the largest masonry dome built ever built. He devised a double shell system to give the dome structural depth while also creating a herringbone bond pattern for the brick so that it could be built without scaffolding (which would have required more wood than existed in Tuscany). He also invented special pulleys, lifts, cranes, and safety devices that were so far ahead of their time that da Vinci would later study them.

In Chicago in the late 19th century you see the emergence of architecture firms that are often headed by a structural engineer and architect pair such as Burnham and Root or Adler and Sullivan (this is where Frank Lloyd Wright started, he also did some innovative structural work like his Imperial Hotel in Tokyo which survived a massive earthquake right after construction completed). In modern times there are interesting combinations like Calatrava, Felix Candella, or Luigi Nervi who are both architect and structural engineer or firms that are some combination like SOM which is an architecture firm that has an integrated engineering department or Arup which is an engineering firm with a design focus.

Architectural Theory: Volume I - An Anthology from Vitruvius to 1870 by Harry Mallgrave

Edit: sources and some wording.

What exactly makes you think 1800s tech people were all scientists and inventors? A scientist is someone who uses scientific methods in order to acquire knowledge. An engineer on the other hand is someone who applies scientific knowledge in order to design/build something of use. Because of this distinction, I think someone performing engineering would have ever been called a scientist. For example, I don't think someone who designed bridges would have ever been called a "scientist".

An inventor is kind of its own thing - it essentially means someone who designs something which is novel. But most things that are designed/built are not novel (such as bridges and structures). So again, I doubt most people performing engineering in the 1800's were being called "inventors"

I'll follow this up with a slightly narrower question; did 12th century China have a formal school for siege engineers? If so, how was it paid for?

Related question: My great grandfather was an "engineer", but at that point it meant he drove a train. Was there a direct path from that type of engineer to a modern engineer?