StockFish - at least as implemented on chess.com - has the habit of favoring repeat moves to see if you are willing to make a mistake that allows a faster(?) or safer(?) win. As SF is in control of the game we don't know how and when it interfaces to syzygy. It might have retained some of its human "I know better" character
SF as implemented on chess.com doesn't appear to interface to Syzygy at all (try practice against computer in a KNNKP White to win in > 36 position).
Otherwise SF does still search once it reaches a tablebase position when Syzygy is connected, but that's mainly so it doesn't take 108 moves over a mate in 2 and look stupid. It limits the search to winning moves and I don't believe it will repeat positions once it reaches the tablebase.
Before it reaches Syzygy, or without Syzygy, it will repeat moves whether it thinks it is winning or losing, but that's only if the repeated position evaluates better than its alternatives; I don't think it's an indication of sentience creeping into the Superficial Intelligence. It has bail out code to avoid a triple repetition if it thinks it's winning.
The chess.com GUI is flawed in that respect. If you try letting it play from move 35 onwards in the variation in the example I posted earlier (below), which is mate in 16 at that point, it draws in 1 move because of the flaw. The play shown is against Tarrasch/SF15. It also draws, but in 10 moves. That's not a flaw in Tarrasch though, it's just that Stockfish can't play the KPvK endgame.
Syzygy uses only the FEN without ply count, move number and castling rights to rank it's moves. It can't be said to recommend a move in positions where a top ranked move has not been selected for a winning side in all positions from the ply count 0 position in the endgame. But the Tarrasch/SF15 play above illustrates that SF may ignore the Syzygy ranking if it has information on preceding play in that case. (The Syzygy top ranked move after move 34 would also draw in 1).
@Zagazik
I believe the DTM figures from that server are simply taken from a DTM tablebase, they're not in fact Syzygy. The 7 man DTM figures disappeared from the reports when some sac of smegma infected the Lomonosov site with ransomware and if you try a 6 or 7 man position with a lone king they're also absent because I don't think Nalimov et al bothered to generate them.
DTM50 tablebases are under construction but don't hold your breath. The figures from those won't replicate the DTM figures though and still won't take the triple repetition rule into account.