Indeed, NASM's optimization choices are inconsistent, assuming that ss and ds are interchangeable (i.e. a flat memory model) when splitting [ebp*2] into [ebp+ebp] to save 3 bytes (disp32 vs. disp8), but not optimizing [ebp + esi] into [esi + ebp] to avoid a disp8.
(And the NASM manual even mentions the different default segment, contradicting the conclusion you drew from the wrong info you got about [0 + ebp*2] vs. [0+ebp+ebp*1].)
EBP or ESP as a base register imply SS, otherwise the default is DS. When two registers are used in a NASM addressing mode, the first one is the base, unless you write [ebp*1 + esi], explicitly applying the scale factor to the first one. An index register never implies a segment, which makes sense if you think about the design intent: an index relative to a segment:offset given by a base register or an absolute disp32.
As written, [ebp*2] is an indexed addressing mode, implicitly requiring 4 bytes of zeros as a 32-bit displacements. You can get NASM to encode it that way with [nosplit ebp*2].
Perhaps NASM and YASM overlooked this corner case, because flat memory models are nearly universal outside of 16-bit code. (And 16-bit addressing modes are different and don't support scale factors. Although you can use 32-bit addressing modes in 16-bit code to take advantage of scale factors and the wider choice of registers, even in pure real mode rather than "unreal" mode which lets you set segment limits high enough that offsets > 2^16 are usable.)
All mainstream 32 and 64-bit x86 OSes use a flat memory model, where SS and DS are interchangeable, making this optimization safe under those OSes when you aren't doing anything weird. Segmentation was sometimes used to make non-executable stacks before that was supported by page tables, but that's still a flat memory model. (64-bit code fixes the base/limit for CS/DS/ES/SS so this optimization is always safe there unless SS is an unusable segment entirely, like maybe write-protected if that's possible.)
Still, any assumption of a flat memory model should be optional. This is a bug in NASM and YASM. They should either respect the difference between SS and DS, or should take full advantage of a flat memory model to help out programmers who don't remember which addressing modes have "hidden" extra bytes required, like optimizing [ebp+esi] with no displacement into [esi+ebp]. Preferably there should be an option or directive to tell the assembler that it can assume SS and DS are the same.
Operands to LEA can always take advantage, because LEA only deals with the offset part of the address so segments are irrelevant. (And this would be the most common use case for an addressing mode like [ebp*2] with no displacement: using that as a memory address would maybe emulate word-addressable memory? That's just weird, normally there's a real pointer as one component of the address.)
Understanding x86 32/64-bit addressing modes:
Other than 64-bit RIP-relative addressing, 32/64-bit addressing modes are any subset of disp0/8/32 + base_reg + idx_reg*1/2/4/8, where each of the 3 terms / components are optional. But at least one of disp32 or base register is required. (See also Referencing the contents of a memory location. (x86 addressing modes)).
[disp32=0 + ebp*2] (with disp32=zero) has default segment = DS. You can get this encoding in NASM from [nosplit ebp*2], and addresses like [ebp*4] can't be split.
[ebp + ebp + disp8=0] has default segment = SS, because EBP is used as a base register.
The encoding that would mean ebp with no displacement actually means disp32 with no base reg, so the disp32 is effectively the base (implying segment register DS, because the base isn't EBP or ESP). This is the case with or without a SIB byte, so [ebp + ebp*1] still has to be encoded with a disp8=0. Other registers don't have that problem, so normally splitting saves 4 bytes instead of just 3 for EBP. (Except for r13 which uses the same ModR/M encoding as RBP, I guess so that part of the decode hardware doesn't need the extra bit from the REX prefix.)
ESP can't be an index register, so [esp*2] is impossible to encode with or without splitting. So the special case of NASM's optimization only affects EBP*2. (base=ESP is the escape code for a SIB byte, and index=ESP in the SIB byte means no index, allowing you to encode [esp + 12].)
But unfortunately NASM/YASM split EBP*2 even when there is a constant that needs a disp32 anyway, like [symbol + ebp*2], where it doesn't save any bytes and in fact hurts performance for LEA (but not loads/stores) on Sandybridge-family CPUs. 3-component lea eax, [symbol + ebp + ebp*1] is slower than 2-component lea eax, [symbol + ebp*2]: higher latency and 1-per-clock throughput instead of 2. According to http://agner.org/optimize/, those would be equally slow on AMD Bulldozer/Ryzen, because a scaled index makes a "slow-LEA" even with only 2 components.
IDK if any old CPUs do better with an unscaled index and 3-component addressing modes, for LEA or for actual memory operands.
NASM and YASM behaviour:
$ nasm -felf32 -g -Fdwarf foo.asm
$ objdump -drwC -Mintel -S foo.o | sed 's/DWORD PTR//'
# (edited to put the NASM source line's addressing mode onto the same line as the disassembler output, instead of separate lines)
00000000 <sym-0x2c>:
0: 8b 04 2e mov eax, [esi+ebp*1] ; [esi+ebp]
3: 8b 44 35 00 mov eax, [ebp+esi*1+0x0] ; [ebp + esi]
7: 8b 04 2e mov eax, [esi+ebp*1] ; [ebp*1 + esi]
a: 8b 44 2d 00 mov eax, [ebp+ebp*1+0x0] ; [ebp*2]
e: 8b 04 6d 00 00 00 00 mov eax, [ebp*2+0x0] ; [nosplit ebp*2]
15: 8b 45 00 mov eax, [ebp+0x0] ; [ebp*1] ; "split" into base=ebp with no SIB byte
18: 8b 04 2d 00 00 00 00 mov eax, [ebp*1+0x0] ; [nosplit ebp*1]
1f: 8b 84 2d d2 04 00 00 mov eax, [ebp+ebp*1+0x4d2] ; [ebp*2 + 1234] ; bad split for LEA, neutral on modern CPUs for load/store
26: 8b 85 15 cd 5b 07 mov eax, [ebp+0x75bcd15] ; [ebp*1 + 123456789]
sym: ; using a symbol reference instead of a numeric constant doesn't change anything
2c: 8b 84 2d 2c 00 00 00 mov eax, [ebp+ebp*1+0x2c] 2f: R_386_32 .text ; [ebp*2 + sym]
33: 8b 84 2d 2c 00 00 00 mov eax, [ebp+ebp*1+0x2c] 36: R_386_32 .text ; [sym + ebp*2]
YASM encodes all these cases identically to NASM.
