While the 80186 has on-board peripherals, they aren't compatible (at least not early 1980s varieties) with the standard PC stuff. As an example, consider the Tandy 2000 or the Mindset (there were others)--they each take their own version of MS-DOS.
But the T2K in particular used very little of the '186s onboard peripherals, if any. Stare at a 2000s mobo sometime. You'll find all of the support chips that are on a 5150s mobo. The biggest reason for many IBM-incompatible's incompatibilities was the usage of say an NEC 7220 graphics chip (I can never get there prefixes right, either pd or upd), or something else entirely. And the graphics memory was in an altogether different location, not to mention the internal architecture, registers and whatnot, being totally different.
The T2K in particular was very much so BIOS and MS-DOS system call compatible. If a program was "well behaved", in the same way that all but the earliest Macintosh s/w was, that is using manufacturer provided interfaces to the hardware, it would run. A few things did run on a Tandy 2000, but not much. Most firms started tapping into the hardware directly, w/assembly language generally, and since that s/w was totally tailored for the 5150/5160 architecture, the pseudo-compatibles were left behind.
If you were to plug certain high performance graphics cards in your 5150/5160, you'd often have the same problems. Vendors like Autodesk would address this w/a patch generally, because architects, engineers and other designers couldn't get by w/CGA graphics. Most people are unaware that there were a bunch of different graphics cards for the peecee, many far superior to anything IBM offered (until the EGA/PGC). But were rarely register or memory compatible w/their lineup.
There are little funky nuances that exist that make the 80186/80188 differ from the 8088/8086, but I don't think nothing that couldn't be solved patches.
The early Ampro Little Board/PC had a 80186 (later ones, like mine, have the NEC V40), and that was for the most part totally compatible w/IBM s/w. Many manufacturers used to use it as a fast 80186, and made use of the onboard peripherals to different extents. There were oodles and oodles of 186/188s used in industrial microcontrollers. I'd like to research them *all*, but I don't even know where to start. On Vets Highway in Bohemia, Long Island, NY, I stopped in to a place for some job related reason when I was about 19 I guess. A guy had an 81086 uController right on his desk. Many many small companies did this.
One of my favorities is the Radio Electronics Magazine RE Robot Brain board (designed and built by Vesta Technology, who still is around). It used an 80188. I have the artwork, rom images, and a chum on this very board who BUILT the whole robot. I'm always so bogged down w/other stuff, I haven't had time to etch my own boards (and I have a new fandangled method of reliably doing this at home). Anyone that wants to get in on this, or building other RE computers, give me a hollar.