The range of the bass never necessitates this tuning complication well known from piano tuning. The "hair" would end up so thin, for a bass, so as to have no effect whatsoever and I have never seen anyone to do this as an explicit tuning step. But it may be implicit in our tuning methods.
I don't find the idea entirely absurd for a fretless plucked instrument with an extra low (and relatively short) lowest string. Am I calling the E string relatively short? Yes I am, it's meant relatively to its thickness. There are two ways to make a string sound extra low. Either you make the string longer, but then the bassists need to grow orchestral player's big hands and perhaps even make uncomfortable detours into their C extensions. Or you alter the weight/stiffness ratio but that makes the string relatively thicker, which is the same thing as making it "relatively shorter".
A string has the shape of a thin cylinder. The thinner it is, the closer it is to a line segment. A line segment has harmonics at integral multiples of the base frequency; a thin cylinder has them a tiny bit further apart of each other. Especially the F2/F1 ratio is affected.
The E string is not normally played in the thumb position because there it becomes a thick enough cylinder that it's no longer perfectly in tune even with its own harmonics (thus sounding a bit muddy). So it usually ends up used only for notes too low to be played on any other string. Its overtones have to be in tune with those higher strings' fundamentals and overtones, while its own fundamental is "free" to be tuned wherever, as long as it is not audibly out of tune with its own overtones (a condition which is only approximately reachable, and not alterable by ordinary tension based tuning). In fact, a common bass tuning technique tunes the 4th harmonic of E to match the 3rd harmonic of A, which would, if executed perfectly, result in the thicker string's fundamental ending up a super-thin hair flat compared to the thinner string's fundamental. Won't that affect all of E, A, D strings equally? Yes it would theoretically all of them, but the thickest cylinder would be affected the most. So if anyone would ever barely perceive or barely measure the effect, it would be probably on the E string.
Won't the same effect be encountered on a cello as well? Well, there are two differences here. One is psychoacoustic. We are better at detecting pitch differences by ear in the cello range than in the contra octave. The fundamental frequency is always physically much "louder" than any of its overtones, but we might have quite a bit more of a tolerance for the fundamental to be a little bit off on the E string (because it is so low), without any such tolerance for the pitch of any of its simultaneously present overtones (also coming from the E string, not just from sympathetic resonance of anything else). The other reason (as Edward has pointed out) is that bowing, as opposed to plucking, provides stable energy transfer to the string, therefore the phase of all the harmonics isn't synchronized among themselves just at the beginning of the tone, but continuously (which is called mode locking). So the harmonics are no longer stretched apart from each other when using the bow, they have to compromise (weighted by energy, thus favoring the fundamental). And the bass gets tuned specifically for pizzicato performance more often than any other instrument in the violin family.
Where does fretlessness come in? Well, any of the above, imperceptible as it is, is sure to drown, on the physical level, in intonation inaccuracies (finger position, finger force angle, action height). But fret positions on a cheap instrument can mess up with intonation even more. So tuning methods which have to rely on anything other than open strings and their overtones will have their own bigger problems to balance than non-integral ratios between harmonics. Tuning is an exercise in compromises, so as to offend the ear the least.
TL;DR - The tuning method matters. When I'm tuning the E string using pizzicato flageolets, or when I'm tuning the bass to match a particular piano, my ear probably locates the pitch of the E string by its overtones more than by its fundamental, and the fundamental does indeed end up a tiny bit flat compared to the fundamental of any other string, while still providing the optimal tuning for subsequent pizzicato play.
