Stockfish has added neural networks for evaluation, much simpler than the large networks in MCTS engines, and efficiently CPU computable. The technique actually originated in a Shogi engine and was later ported to mainline Stockfish, so there's no direct inheritance from AlphaZero in that sense.
The search algorithms have been incrementally improved, but still follow the same Alpha Beta and a heap of heuristics approach.
Stockfish now uses a neural network in some situations. Last I checked, it uses the classical, fast, tactical search and evaluation for active, tactical positions, and uses the network for quieter, more strategic positions.
