Is the native Bitcoin client intended for mining? I can't understand, a native bitcoin client for calculations - will he be able to take part in mining?
If not, what kind of native mining software should use? Does he even exist?
I only found third party products like cgminer, cudaminer, etc.
Is the native Bitcoin client intended for mining?

No.

Long long A decade or so ago, back in the mists of time, in prehistoric days when dinosaurs roamed the earth, early versions of the Bitcoin software did indeed include a mining capability that used the pitiful capabilities of CPUs. But this function has long been rendered defunct by the advance of mining hardware such that one small ASIC is now more powerful at mining than a billion 4.6 million† modern high-end decentish desktop computers even if each were crammed with dozens of overclocked multi-core CPUs and GPUs. So the vestigial and useless appendix was removed from Bitcoin-core's DNA mainnet phenotype.




Hardware
Hashrate Mh/s




Intel® Core™ i7-2600 / 8 / 3.40 GHz
23.9


Antminer S19 Pro
110,000,000.0




Your choice of mining hardware determines what specialised control software you use.
(Health warning: answer may contain microscopic traces of hyperbole)
† Roughly. Give or take an order of magnitude, or two.