Does the KID, Pirc, and Modern opening connect?

Pirc and Modern have several common setups and themes. The KID is a totally different kind of thing.

To expand a bit, the key thing about the KID is that White has committed to c4, which means he can never play c3 to support d4 or block the diagonal against Black's bishop. So White is very rarely castling Q-side, and when Black strikes at d4 with ...c5 or ...e5, White can't leave the pawn there. He either has to exchange it, or (more commonly) advance it. So typically you get K-side castling from White and a locked center, which is what leads to the K-side attack from Black and Q-side expansion from White which is characteristic of the KID, see for example Gelfand - Nakamura 2010 as a classic example.

In the Pirc/Modern, none of this applies, which means White can and often does castle Q-side and throw the K-side at Black, and/or can play c3 to support dark-squared pawn structures in the center, like a c3-d4-e5 wedge against Black's bishop. So you get totally different structures and totally different middlegame plans in these openings compared to the KID.

Pirc, Modern, and Philidor have similar setups. KID is a totally different beast. Modern can sometimes transpose to KID also if white allows.

Slight correction to post 2.

He is totally right that the Pirc and Kings Indian are NOTHING alike. White's c-pawn being on c2 vs c4 makes a HUGE difference.

However, the modern has ties to both openings. What is coined the "e4-modern", where White DOES NOT play c4, like 1.d4 g6 2.e4 Bg7 3.Nc3 d6 etc, with 4.f4 or 4.Be3 or 4.Nf3, there are many similarities to the Pirc.

BUT, the "d4-modern", also known as the Averbakh, after 1.d4 g6 2.c4 Bg7 3.Nc3 d6 4.e4 Nc6 5.Be3 or 4...e5 5.Be3, This has more similarities to the Kings Indian Defense, often lines where Black plays ...exd4 as opposed to those lines with d4-d5 by White.

So more accurate would be the e4-modern (NO c4) and Pirc are similar, the d4-modern (WITH c4) and the Kings Indian have a lot in common, but the 2 pairs do not cross paths with each other and are VASTLY different.