How can I access my second Segwit2X currency, if I own private key, but my wallet does not support Segwit2X currency? Is that private key universal for every wallet? I mean if I can access my BTC or Segwit2X by just my identification with private key in any or some wallets. How does it work? Will my wallet automatically support Segwit2X or what?
To use the SegWit2x chain directly you'll need to download and build the btc1 full node from the git repo. You then copy your existing Bitcoin (SegWit1x) full node wallet file into the btc1 directory to load your existing private keys.
BUT I wouldn't do that. There doesn't seem to be a safe way to spend SegWit2x coins without risking your Bitcoins. It seems like the recommended approach is to wait until after the dust has settled, with some articles indicating this could take a few days after the fork, and then determine if there's a safe way to spend your 2x coins (most likely by converting them for 1x coins which I'm sure is what you had in mind).
Update: You could take a look at the chainspl.it initiative and their associated design doc to get some additional ideas. Although after perusing the site and reading the doc I'd still say it's arguable whether the "easily handle forked coins" tag can be lived up to.