Ttdfjt

The description of "5/8 of the way into [your] meal" could imply that you are:

Eating octopus in a high-end seafood restaurant. 'High end' since they have a strict dress code...

This particular establishment:

Is so high-end that it prides itself on only serving the freshest food - and what better way to ensure the food is ultra-fresh than by the restaurant being actually at the quayside in a port town?

This means that among the clientele would be:

Sailors and other seafaring men and women from the boats which dock nearby.

The shoe-less customer could therefore be:

A sailor with two wooden legs!

(Similarities with SteveV's answer perhaps but hopefully you can also see the differences and the justification...)

It's additive

No shirt no shoes no service.



No shirt, shoes, half-service (1/2 a meal).


Shirt, no shoes, half service.


Shirt, shoes, full service.

therefore,

Taking shoes off is forbidden once you've had more than half of your meal.

I think it is because

one of the people had no legs, perhaps in a wheelchair.

If so

you should have known better!

Alternatively

perhaps one was a baby

That would explain the waitress's laugh, but still not the hint.

The hot summer could be a thing anywhere but I want to assume you were

in South or South-East Asia

Now I don’t know what exact type of restaurant it was or what your food was. However, I want to guess that the shoeless person

was also wearing no shirt but rather a Kasaya or a robe because they are a Buddhist monk or Bhikku.

Of course, when you pointed that out to the (local) waitress

she would try to friendly laugh off the potentially insulting comment, which you misinterpreted because you were on holiday.

Obviously, you cannot take your shoes off in this context but when you did and got into an argument

the waiting staff informed you that Buddhist monks are well-respected people who have a traditional reason to be with or without shoes at any time. You are of course not one and attempting to put yourself on their level is rude and impersonating. And because you obviously don’t (want to) understand this, the troublesome customer is ultimately kicked out.

The significance of the hint:

A key idea of Buddhism is the Noble Eightfold Path which is often symbolised by a wheel with eight spokes; this is represented by the 8.

The fifth point of the eight is given on Wikipedia as ‘Right Livelihood: beg to feed, only possessing what is essential to sustain life’ which connects to the fact that you are in a restaurant, hence the five. Only one of the two people is a monk so I think it is possible for the other one to donating food in that way but that may be a stretch of my imagination.

I guess I’m completely wrong but it’s a little bit lateral and I like the way I incorporated the hint.

The reason is that:

It’s a pizza restaurant and the other customers were getting a takeaway. The rules only apply to customers who are staying in the restaurant to eat their meal.

Motivation:

5/8 is oddly specific - unless you’re eating slices of pizza. Then it makes sense to quantify how much you’ve eaten this way.

Each party needed shirt and shoes, not each individual. You did not have shoes on, which is why you were refused service. In the party of 2, one of them fulfilled the shoes requirement, hence they were allowed to receive service.

Perhaps this is a specific

Eight course meal with entertainment

Where

The sixth course is often served with a light sorbet (or similar) to cleanse the palette and where there is entertainment offered. The two people who came in were the entertainment. The entertainment was a ventriloquist with a dummy. "They" came in and sat down as the entertainment. The dummy wasn't wearing shoes and talking about that "dummy not wearing shoes" could be misconstrued and cause an argument.

There is a sign outside which reads "no shirt, no shoes, no service."

But

Thats only valid for [people of group A].
Maybe for [people of group B] there is another sign beneath which reads "no shirt, no shoes, service."

A hint to this maybe the title of the question.

So the shoeless person also has no shirt and is a [people of group B] and qualifies for service per the second sign.

Well, and 5/8

maybe the percentage of people who care about my last edit.

An incorrect solution:

One of the people got inside with the shoes, then passed the shoes to the other person for them to also enter with shoes.


It's wrong because then the waitress would have kicked them out for not having shoes on.

A plausible solution:

Since they're a group, only one person needs to have shoes. (nowhere does it say that each person needs to have shoes, only that you need shoes)


It's probably not the correct solution because the number 5/8 doesn't play a role

At this restaurant,

if you have no shirt and no shoes, you will get no service. In other words, you need a shirt and/or shoes to get service.

Because it was a hot summer day,

you were wearing no shirt and no shoes. When you got your shoes, you qualified.

The other customer had a shirt on, so it didn't matter whether they had shoes on.

And I suppose

the ⅝ thing is a red herring?

One of them had

sandals on, technically not shoes