Apple Watch series 6 shows weird charging behavior with series 2 charging cable

I agree with @ckuan, the cables are common for all watch series and should just work. Although it is possible both older cables are defective it may also be a problem with the power adaptor, or even a poor connection between the cable and the watch (give the back of the watch and magnetic pucks a wipe over with a damp cloth). Finally, test by substitution with the cable which came with your Series 6.

You say it is obvious, but don't say why you think that or what you have actually tested. All Series 6 are supplied with a new charging cable, did you test and confirm the fault also occurs with the new cable? Did you test with a different power adaptor on any or all of the cables? Did you even test with power from a different wall socket? These are all things you have never mentioned but they are easy to make a quick check without the effort of going to an Apple Store (probably closed or reduced access with the pandemic) or mailing your watch to a Service Center.

If you do take/send your watch for inspection it would be good to send the charging accessories with it so they can test the complete setup.

And with that you have probably proved the fault. A Series 6 is only a few months old so it is certainly still in warranty cover. The only extra advice I can offer at this stage is to grab a few photos as you attempt the unpair/pair process, simply so you can prove to the disbelieving support phone jockey that you have already done points 1-8 on the troubleshooting checklist and they should go direct to a warranty replacement.

As far as arranging service is concerned arrangements may vary around the world particularly with the pandemic. In general it is best to start online with (link) Get Support and setup an Apple Store appointment, or if possible arrange a replacement/exchange device mailed out to you.

Yes, definitely a problem for Apple Service.

Your link to the pic and videos shows me your real name, which you might not want to share with the world for ever more. I'll report it and ask our moderators to redact the link for your privacy

I am beginning to think this is a problem with our OLD USB-A adapters. New apple watches and iPhones ship with the cord, but no adapters. I bought an iPhone 12 Pro Max and an Apple Watch series 6 on the same day last weekend. The iPhone, of course, had a USB-C charging cable, but no adapter. I bought another USB-C adapter to have for the phone. Unpacked everything, and ODDLY, given Apple's clear direction, the Apple Watch MagSafe charger had a USB-A connector. All of our USB-A connectors are 5+ years old. Not sure if that was a factor, but I COULD NOT get the Apple Watch to charge. Pairing with iPhone was smooth as silk. Then got prompt to update Apple Watch software, but could not because it require watch to be plugged-in to charger and at least 50% charged. I could not achieve that goal. Apple Watch never recognized it was plugged in, and though it charged briefly, that was it, never got above 30% charge. Not sure where the weak spot was. Could be my OLD Apple 5w adapter that still works great for OLD iPhones. Why did NEW Apple Watch ship with USB-A cable but NO ADAPTER in 2021 when Apple's direction has been USB-C since 2017? They are forcing us to use ancient stuff or buy NEW obsolete stuff. AND THEY HAVE STOPPED PUBLISHING PRODUCT REVIEWS. I will return the watch, thank you.

Maybe your cable is defective.

I have no issue using cable from an Watch Series 2 to charge Watch Series 4, 5 and 6

Yes very unlikely.

FWIW: I also have 2 Series 2 cables from my kids and they both have upgraded to Watch Series 6.

The 2 old cables are still been used to charge their Series 6, it's working fine.