This is the 184th most frequent Polish word.
samym
"Samym" means "the very" or "oneself" depending on context.
Here 'samym' emphasizes the identity of the problem, being exactly identical to another referenced issue.
On poradził sobie z tym samym problemem.
He dealt with the exact same problem.
In this sentence, 'samym' is used for emphasis, meaning 'exactly', highlighting the timing occurrence.
Przyszedł do nas pies w samym środku nocy.
A dog came to us in the middle of the night.
The word 'samym' is used here to indicate 'the same'. It serves as a modifier to designate matching attributes (time in this case).