Is 13 years too old for an otherwise new-condition tire?

A 13 years old not used rubber is always subject to degradation of properties and it doesn't matter how good it is stored. Its operation time will be quite lower than a new produced tyre and there is no way to understand when and where it will fail.

If manufacturers find such products in their warehouses, they would scrap them for sure and investigate how could they stay like that for years.

My only doubt is its production time format, I have seen so many different date formats on rubbers, batteries, pipes, hoses etc. Most probably you read it correct but it doesn't have to be Day/Month/Year all the time. It could be serial number/quarter/year, pp/year, week/year, year/week, only last two digits of the year or all 4 digits of the year, year/batch number, etc... My point is 07122017 could be any of these:
^{date is for example}

• batch 0712, year 2020, week 17
• batch 0712, PP 20, year 2017
• or just 07 December 2017

is there a valid concern about otherwise new-condition tires that are 13 years old being unfit for use?

Yes, it is a valid concern.

As user48539 says tires degrade no matter what.
What we would like to know is:

• Why?
• How old is too old

Why?
The answer to "why" is oxidation over time.

I could find no studies that focus on bicycle tires specifically. The following is an excerpt related to automotive tires. Similar forces are at work on bicycle tires.

The phenomenon of tire age degradation or thermo-oxidative aging has been documented in technical papers going back more than 80 years, and according to one journal article from 1931, it wasn’t news then: “The auto-oxidation of rubber has been known for a long time, and for a long time, too, it has been known that it plays an important part in spontaneous deterioration or aging, and it has been the object of numerous studies of much interest.” Safety Research & Strategies Inc

Bicycle tires will oxidize at different rates depending on many variables. Exposure to sun, higher temperatures, the type of materials used in the tire, tire coatings, etc. can have an effect on the rate of oxidation over time.

What I've seen is that age - oxidation - reduces the flexibility of the materials that make up the bicycle tire.

As a tire rolls it flexes where it comes in contact with the road. When the casing material becomes inflexible and breaks it can't hold the bead and the tube will blow out of the tire. When the tread becomes inflexible it wears faster.

The tire tread, casing and sidewall become "crunchy" over time. They begin to break rather than flex.

How old is too old?
It depends on all those variables mentioned in the "why" part of the answer plus it depends on the application for the tire. Higher pressure, higher performance tires are less tolerant to degradation. And generally, higher pressure / performance tires are more likely to hurt the rider when they fail. If you have a low pressure, low performance application sometimes you can get away with a little more degradation.

You can tell a lot about a tire by how it feels.
(I'll probably catch some heat for this but here goes)
Get the tire in your hands and compare how it flexes to a newer tire. If you can twist the tire into a figure 8 with the same effort as a new tire, and with no crunching sounds, especially for a low pressure, low performance application - it's probably OK.

This won't help with mail order. Best to avoid old tires via mail order.

A 13-year-old tire may be rideable for a few years, but it wouldn't be fair to sell it as new. It just won't last as long as a new one would, and the hardened rubber will perform worse and possibly wear faster.

So, not necessarily "unfit for use", but definitely "unfit for sale as a new one". But depending on storage conditions and the tire, even a few years can make a tire unusable, such as obvious hardening or visible cracks.

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As for the particular review linked in the question, it seems the buyer has misread the date code.

The page identifies the tires as Schwalbe Marathon HS. The Schwalbe Warranty Page has this photo of their production code:

The format appears to match the one in the question, but it still leaves at least two reasonable interpretations:

• Day/Month/Year: giving dates of 20th July 2018 (question) and 26th September 2018 (warranty page).
• Week/Year: giving dates of week 20, 2007 and week 26, 2009.

But, it appears that Schwalbe has changed their production code format with an extra digit in 2010 or so. Older sources, such as this recall from 2009 show a code that has only 5 digits before a letter:

Based on this, I'd say the format is Day/Month/Year and the date on the tire is 20th July 2018. It still makes the tires somewhat old to be sold as new, but not terribly so.

All polymers degrade over time. If they didn't, it would be hard to quickly produce them in the first place (chemistry, that is).

Then again, the speed of the degradation can be orders of magnitude slower or faster, depending on the conditions. A tyre stored in cold, dry and dark place for 15 years can be way better than a tyre exposed to the weather conditions for a few months.

An adequate rider quickly adapts to tyre's age-reduced performance without even knowing that, just like one adapts to safely handle different brands and types of tyre.

I am sure I have used both automobile and bicycle tires older than 15 years with no ill effects, just by not doing stupid things.

The only inevitable result of an aged tyre are more frequent flats - up to the point of becoming unpractical.

It totally depends on how the tires are stored, back in 2018 I received my grandmothers old Phillips road bike still rocking it’s original Dunlop roadsters from 1956 they were not dry rotted or anything of that nature and I was able to squeeze two years of almost daily use out of them before the sidewall on the front tire started to became bloated enough it was hitting off the frame. I would have replaced them immediately but the tire sizes here in North America are different than that of Europe so getting a hold of new ones was a challenge