Verified answer

No, it is just average phrase. What it means, it depends on context.

In general it means:

A trick/thing made by Russians which can be managed/controlled/etc... only by Russians.

For example:

This Russian car is a thing which can be driven only by Russians.

Эта русская машина - штука, подвластная только тем же русским.

Behaviour of this Russian is a trick which can be understood only by Russians.

Понять поведение этого русского - штука, подвластная только тем же русским.

that mean "thicker, independence day greetings only the same russian"