Tangible is simply not a common term
In fact, no spells in the PHB say that the spell effects they produce are 'tangible'. Two spells specifically say that their spell effects are intangible: the illusory copy of project image, and the scrying sensor of *clairvoyance 1. Clairvoyance says (emphasis mine):
A creature that can see the sensor (such as a creature benefiting from see invisibility or truesight) sees a luminous, intangible orb about the size of your fist.
While scrying says:
A creature that can see invisible objects sees the sensor as a luminous orb about the size of your fist.
It would be foolish to conclude that the scrying sensor of scrying was tangible simply because its description does not say that it is intangible, when intangible is used in the description of only two spells. We need to look for other terms.
Can things pass through the image?
Some illusion school spells create effects only in the mind of the target, while others create objective external images that can be seen by any observer. For spells of the latter kind, often times these images can be detected as illusions because they "fail to hold up to physical inspection" and "things pass through" them. This is the case with minor illusion, disguise self, silent image, major image, hallucinatory terrain (where 'the tactile characteristics are unchanged'), seeming, programmed illusion, and project image.
For illusion spells that create external images, these images typically appear 'at a spot within range'. Notably, they do not have the restriction of having to appear at an unoccupied spot within range such as spells that summon creatures or sizable solid objects do (including other illusion spells). The clear implication is that illusory images can appear in occupied spaces, and thus can co-occur with solid objects, because they can pass unhindered through the objects and creatures in the spaces of their appearance. See silent image, mirror image (where the space is explicitly occupied by you), Nathair's mischief (where the space explicitly may be occupied by creatures), major image, mislead itself (where the space is explicitly occupied by you), programmed illusion, and project image.
Can you feel it?
Some illusion spells do create images with tactile components, although they are fewer than the ones that expressly do not. We know they can be felt because the spells explicitly say so. These include shadow blade, phantom steed (which must be summoned to an unoccupied space), creation (where the illusion's properties explicitly include its hardness and density), mirage arcane, simulacrum, and illusory dragon (which must be summoned to an unoccupied space). Many, but not all, of these spells reference gathering material from the Shadowfell to justify the ability to create solid illusions2. The three that create creatures rather than objects or terrain (phantom steed, simulacrum, illusory dragon) reference their AC and hp. It is safe to say that if illusions are tangible they should have spell descriptions that indicate them as such.
Evaluating mislead
Mislead does not say whether the illusory double is tangible or not, which tells us nothing. It does say that it can be summoned in an occupied space, which is evidence for it being intangible. It does not reference the double as being able to be felt, or having to draw on the Shadowfell to make it, or having an AC and hp, all evidence for it being intangible. Strangely, for a externally objective intangible image spell, it does not tell us that it can be detected as an illusion by touch. This is so anomalous I have to believe that it was an editorial oversight where the 'fourth paragraph' about detection was simply left out.
We don't have definitive proof that the misleading double is intangible, but the evidence strongly suggests it.
1 The term 'intangibles' is also used in the description of the imprisonment spell, but in a metaphorical sense.
2 Most likely ultimately a callback to first edition lore where illusionists could create quasi-real effects through the specialized process of demi-shadow magic and shadow magic.