[RPG] Is the Cloak of Many Fashions ability an illusion or an actual physical change

Strictly better? No.

One of the major points of Disguise Self is that it can not only alter your appearance (via illusion), but your clothing and equipment as well.

It's important to note the inverse of this: Alter Self does not specify that it modifies your clothing or equipment. This means that, depending on how severe your alterations are, you may no longer fit into your armor and your clothing may clearly no longer fit you, depending on how simulationist your DM is on the topic. This is especially important if using the "Variant: Equipment Sizes" rule on PHB p.144:

In most campaigns, you can use or wear any equipment that you find on your adventures, within the bounds of common sense. For example, a burly half-orc won’t fit in a halfling’s leather armor, and a gnome would be swallowed up in a cloud giant’s elegant robe.

The DM can impose more realism. For example, a suit of plate armor made for one human might not fit another one without significant alterations, and a guard’s uniform might be visibly ill-fitting when an adventurer tries to wear it as a disguise.

Using this variant, when adventurers find armor, clothing, and similar items that are made to be worn, they might need to visit an armorsmith, tailor, leatherworker, or similar expert to make the item wearable. The cost for such work varies from 10 to 40 percent of the market price of the item. The DM can either roll 1d4 × 10 or determine the increase in cost based on the extent of the alterations required.

Even if your equipment does fit you after the effects of the spell, any well known gear or accessories may still let you be identified, especially if they're more well known then your actual physical attributes.

There are a few other considerations as well.

One of them is that both Alter Self and Disguise Self can be used at the same time, to enhance the illusion with actual physical changes underneath it. Do note that this is only possible because Disguise Self does not require concentration, which is another point in its favor over Alter Self, which does. This means Alter Self can fail earlier than intended (especially in combat or while under stress), while Disguise Self will last the duration under most conditions, and still allows you to cast other concentration spells.

Another is comparing how Disguise Self and Alter Self interact with height & size.

Alter Self actually changes your height, while Disguise Self only creates the illusion of a height change, keeping you your usual size. This might matter if your DM actually cares about character height in certain situations rather than just your size.

In addition, Alter Self restricts you to only changing your appearance to a creature of the same size with no other restriction on maximum height gain/loss, while Disguise Self only has a restriction on height change rather than size change.

For example, with Alter Self, you could change the shortest possible dwarf into the tallest possible goliath, whereas with Disguise Self, you could make a dwarf appear to be a halfling (so long as you're not trying to change your height by more than a foot in doing so).

Overall, I'd probably rate Alter Self better than Disguise Self in some combination of the following situations:

• There is plenty of time to make preparations in advance, including having an appropriate alternate outfit ready.
• You are unprepared, but you only need minor superficial changes with no drastic change such as size, making the lack of alternate costume not an issue.
• You are in no danger of having your concentration broken for the next hour, nor are you likely to need to cast any other Concentration spells.
• You are likely to be thoroughly inspected for some reason or another.

In other situations, there is a chance Disguise Self may be the more appropriate spell. In a situation requiring an emergency drastic appearance change (clothes, size, and all), for example, I would much prefer Disguise Self as my option.

Minor Illusion and Major Image have the following paragraph (slight differences in the last sentence, to include the additional features of Major Image.

Physical interaction with the image reveals it to be an illusion, because things can pass through it. A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the illusion becomes faint to the creature.

So, there are two ways to look through an illusion:

1. If anything physically interacts with the image, it is revealed as an Illusion (automatically)
2. If a creature uses its action and succeeds on a Intelligence (Investigation) check

What are fair ways for the GM to interact with those Illusions?

The core should be, that the GM does not use his meta knowledge.

That means if those illusions are cast out of the enemies sight the enemies would interact with it as if the illusion is real. You could use an illusory wall as full cover and peek around to attack your enemies, but as soon as you attack through the wall, the enemies will know that something is up. (physical interaction)

An exceptional case would be "what happens if you have three-quarters cover behind an illusion?" the core principle would be, that the enemy would try to avoid your cover and thus you could benefit from the cover bonus, but what happens if he misses? Does he hit your cover and reveal it as an illusion? That is up to your DM to decide.

(I personally would say, if the attack would hit if there were no cover bonus, the target hits the illusion and reveals it as such, but one can argue that the attack should have hit you then, too.)

If you happen to cast the spell while your enemies can see it there should be a few differentiations:

1. an enemy who is stupid (beasts, low-int creatures)
2. an enemy who is intelligent, but has little to no experience with magic (maybe bandits, barbarian tribes, more intelligent beasts)
3. enemies who are intelligent and are either able to cast those spells themselves or at least know they exist

The first group should interact with an illusion as if it is true and avoid it.

The second group may use their action to determine if it is an illusion and tell their comrades of their results and then react accordingly.

The third group would know that this is probably an illusion and ignore it. Attack your spot based on position or just by listening (attack unseen target). The only reason for them to look closer to the illusion would be if they need to see their target for their spells. And as soon as the first enemy shoots through the illusion it would reveal itself.