I'm wondering where the so-called Intel syntax for x86 assembly came from. Did Intel release their own assembler for chips like the 8086 or do we just mean the syntax they used in the manuals?
Yes, Intel did distribute an assembler using the syntax described in their manuals. The official 8086 family manual states (§1.3, p. 1-12):
Intel provides the sophisticated tools needed for timely and economical development of products based on the 8086 family. […]
Three language translators support 8086 family programming. PL/M-86 is a high-level language for the 8086 and 8088 that supports structured programming techniques. […] ASM-86 may be used to write assembly language programs for the 8086 and the 8088 CPUs and gives the programmer access to the full power of these CPUs. 8089 programs are written in ASM-89, the 8089 assembly language.
However, for all I know the term ‘Intel syntax’ did not exist until its referent had to be contrasted with the AT&T syntax, and that is what the term is primarily used for in practice: it just means ‘x86 assembly syntax that doesn’t modify mnemonics, doesn’t require sigils and puts the destination operand first’, without necessarily implying 100% compatibility with syntax found in official Intel manuals.
I'm wondering where so-called
Why so-called? How else to call it?
Intel syntax for x86 assembly came from.
Intel developed it.
It has been a steady growth. Basic mnemonics and structure are a continued development since 8008, which itself is based on the Datapoint 2200 Assembler. See the history part of this Answer for some remarks about the lineage (Datapoint -> 8008 -> 8080 -> 8086).
For a more detailed answer it might be helpful to know what specific elements poke your interested.
Did Intel release their own assembler for chips like the 8086 or do we just mean the syntax they used in the manuals?
Intel did, like any other CPU manufacturer/designer, release their own assemblers - in fact not only assemblers but whole development systems, which at that time meant dedicated workstations, not just software packages :)).
To the best of my recollection, 8086 assembly language is an upgrade of 8080/8085 assembler expanded to use the new features and registers of the 8086. Also, I half remember talk that 8080 assembly programs could be reassembled for the 8086. (There may have even been three hobbyists in Nebraska who actually used that design feature. Of course, those were the grandkids of the three grandmas who bought the original IBM PC to use with cassette interfaces on their 64 kbyte machines...)