What is depth (plies)? Does a fast-climbing depth mean strength?
For today's strong engines, depth (plies) is no longer a true "depth". Depth (plies) has almost no connection to how many moves ahead the engine calculates, because modern pruning and extension strategies make the search depth of different moves very uneven.
The depth displayed now only represents how many times the search has completed a full iteration of its search steps. Some moves are searched deeper than this number, and most moves are searched shallower. If there is a forced mate, a repetition, or the position is approaching the 60-move rule, the amount of computation is small and iterations go very fast, so you may often see depths like 245.
The word "depth" has been kept only for historical reasons.
So depth n does not mean the best play within n moves; the two are almost unrelated. Even if the search reaches dozens of moves ahead in some lines, in extreme positions it may fail to see a mate in just a few moves.
How fast depth climbs does not represent playing strength. The only indicator of playing strength is scientific test data. Fundamentally, the longer the engine computes, the stronger it is.
The speed at which depth climbs may differ between positions, between thread counts, and even by luck. At the same so-called "depth", more threads are generally stronger; it is not the case that the same engine at the same depth always has the same strength (that only holds with 1 thread).
With multiple threads, the main gain is in search-tree width, and the speed at which depth climbs can be seen as a by-product; sometimes it may even be lower than with fewer threads. Looking only at how fast depth climbs is wrong.
For the same engine, regardless of instruction set, and on different machines regardless of hyper-threading, as long as the settings that can affect the search path are identical (such as hash size, rules settings, multipv) and the hash has been cleared, the 1-thread analysis at each depth is identical; only nps and time differ (multithreading has randomness). So with a single thread, a larger depth does reflect strength, because it means the search ran longer. With multiple threads, however, such "depth" can no longer be used to compare engine strength: multithreaded depths can be compared neither with each other nor with single-threaded ones.
For ordinary users, if depth climbs noticeably slower than earlier in the game, it may mean the engine has reached some critical variations that reduce pruning and so slow it down, or that the later position is so lopsided that the scores no longer differ noticeably. Beyond that, most of its significance is just that it looks nice.