So for n steps, one player can only choose one out of many moves, because all others lead to forced checkmate sequences. Only one path from the root to the leaves of the game tree avoids a certain loss.
Does this concept have a name? Are there any known positions where this is the case?
To start off, with such a requirement, the sequence would have to end in a draw. Well, more than most likely that is.
There is a similar concept from another question called the "Nunn Convention." It is a term to describe a move by one side that is the only way to draw, with the extra requirement that it isn't the only legal move. That extra requirement is what makes it "Nunn." It isn't quite what you wish for, but it is close.
The Matplus.net thread from which I learned the concept contains the below example. It is the record length of 32 plies for such a "Nunn" sequence. It is from post number 19.
[Title "Geir Sune Tallaksen Østmoe, Matplus.net Forum 8/9/2019, White To Draw"]
[FEN "k1B5/r1p1p2b/nPPP2R1/BN2b1R1/2P2qp1/p7/P3P2P/6Kn w - - 0 1"]
1. Bb7+ Rxb7 2. Rg8+ Bxg8 3. Rxg8+ Nb8 4. Rxb8+ Kxb8 5. bxc7+ Rxc7 6. dxc7+ Bxc7 7. Bxc7+ Qxc7 8. Nxc7 Kxc7 9. Kxh1 Kxc6 10. Kg2 Kc5 11. Kg3 Kxc4 12. Kxg4 Kc3 13. h4 Kb2 14. h5 Kxa2 15. h6 Kb1 16. h7 a2
The sequence can be arbitrarily long if we allow repeated moves, e.g.
[Title "Every other White move allows mate in 1"]
[FEN "2B5/8/R7/4rp2/2P5/1PkP4/2p6/2K6 w - - 0 0"]
1. Re6 Ra5 2. Ra6 Re5 3. Re6 etc.
Naturally if the "certain loss" incurred by a deviation is allowed to take longer than mate in 1, there are simpler positions, even with only one Black pawn in addition to the two Kings; for example in the following position White must "find" 10 only moves before the first repeat, though it might take Black as long as 26 moves to punish a misstep:
[Title "Every other White move loses"]
[FEN "8/8/k5p1/8/8/8/K7/8 w - - 0 0"]
1. Kb2 Kb6 2. Kc2 Kc6 3. Kd2 Kd6 4. Ke2 Ke6 5. Kf2 Kf6 6. Kg2 Kf5 7. Kf3 Kg5 8. Kg3 Kh5 9. Kh3 Kg5 (9... g5 10. Kg3) 10. Kg3 etc.
I'm assuming that only one side's moves must be unique in this sense; if you want both White and Black to be walking such a tightrope at once, see Rewan Demontay's example of a "Nunn sequence".
I found this position, White has to make two moves (e3, e4) to avoid getting checkmated by Blacks pawn promotion and then has to move to h6 to avoid a draw by stalemate. After these three moves white wins by its own pawn promotion.
[Title "depth 3, white to move"]
[FEN "k7/Pp4p1/1P6/3p3P/8/1p2p3/pP3P2/K7 w - - 0 1"]
The moves themselves are technically forced, so that might be the concept to which you are referring (forced moves).
