Note that there is separate "program RAM" and "data RAM", the latter having a more complicated structure with the aforementioned status characters. It's surprisingly complicated to address any RAM on the 4004, with both types of RAM having separate quirks for doing so.
There are also plenty of other oddities, like I/O ports on both RAM and ROM chips, dedicated instructions for e.g. "keyboard processing" but no AND or OR instruction (with horrifyingly long examples on how to implement them in this document), and some instruction names for seemingly straightforward things that are completely unintuitive.
It's well worth exploring.
It makes sense, AFAIK it was designed for calculators.
The origin of x86 is the Datapoint 2200, a desktop computer built from TTL chips. Datapoint contracted Intel to build a single-chip version of the 2200's processor, and Intel built the 8008, which was essentially a clone. Datapoint decided not to use the chip and gave the rights to Intel and the rest is history. Intel improved the 8008 to get the popular 8080 microprocessor. Intel then created the 8086, which was a 16-bit processor loosely based on the 8080. From that, the x86 architecture developed. Many features of x86 can be mapped back to the Datapoint 2200, such as little-endian.
Perhaps you are confusing it with the 8008 which is related to the 8080 which is related to the 8086 which is related to the 80286, 80386, 80486, and so forth.
Also a lot of instructions have similar names: CLC
You are right, it is not a true ancestor, but ready through the manual, it did seem to me that it was designed by the same people.
To be honest I haven't dig into other CPU families, so maybe these were pretty common.
But I can definitely see 4004 DNA in x86.
One thing the 4004 and 8008 do have in common is a similar floor plan for the chips. They both have the ALU on the left, data bus along the top, instruction decoding in the middle, registers on the right, and on-chip stack on the right. Other processors of that era (e.g. 6800, 6502, 8080) have very different layouts so it's not just coincidence. Texas Instruments produced the TMC 1795, a chip equivalent to the 8008 (before the 8008), and it also has a completely different layout.
If you look lower-level, at e.g. the ALU implementation or the register implementation or the pin drivers, there's less similarity. I would have expected they would essentially cut-and-paste some of these low-level pieces, but they are more different than I would expect.
The 4004 has all kinds of strange instructions, like DAA (for BCD) and separate instructions for accessing main memory, ROM, and RAM. It's pretty clear that it was made to be in a calculator.
On the other hand, the 8008 has the familiar IN, OUT, MOV, PUSH, etc.
9N GHI Get high reg N R(N).1->D
The "Break and Make Smoke" instruction was the predecessor of the "Halt and Catch Fire" instruction.