Java 1.8 and the servers that stayed there
In 1.8 there is no attack cooldown. Every left click on a target is a hit, damage is applied, and knockback is applied. Clicking faster means more hits per second and, more importantly, more knockback events, which is what pushes the other player around and keeps them off you. This is why 1.8 PvP scenes cared so much about CPS, and why the Kohi test exists. Beyond roughly 10 CPS the extra hits mostly get eaten by the game's damage immunity ticks; the reason to go higher was knockback and the mind games around it.
Most of the big PvP networks run 1.8 combat rules to this day, on any client version. If you play on one of them, CPS is part of the game.
Java 1.9 and later
1.9 introduced the attack cooldown. A swing before the cooldown bar refills does a fraction of the damage, and for a sword the full recharge takes about 0.6 seconds. Spam clicking a sword in vanilla 1.9+ is worse than clicking once every 0.6 seconds. CPS is irrelevant there, and timing is everything.
Where clicking speed still shows up in modern Java: placing blocks, bridging, spam-clicking through a crowd of mobs with a weapon you don't care about, and any server that has switched the cooldown off with a plugin, which many have.
Bedrock
Bedrock Edition never got the attack cooldown. Combat there works like 1.8: faster clicking, more hits. On touch screens the game auto-attacks while you hold, which changes the question entirely; on a phone the tap counter is the closer test.
What servers allow
Anti-cheat plugins don't measure CPS the way this site does, but they all watch the click rate. Common thresholds people report: warnings around 15 to 16 CPS, kicks or flags somewhere between 20 and 25, instant flags for perfectly regular clicking at any speed. Drag clicking trips the first two routinely. Double-clicking mice, whether worn or bought for it, are banned outright on many networks. Some servers set the bar lower and say so in their rules; that page is the one to read before you train for a number.
What to train, then
- 1.8-style servers: a steady 8 to 12 CPS you can hold for a whole fight beats a 16 that lasts two seconds. The Kohi test with its second-half number is the right drill.
- Vanilla 1.9+: don't. Practise the cooldown rhythm instead; the reaction time test is the more relevant number.
- Bedrock: as 1.8, with the caveat that touch controls do a lot of it for you.
None of the tests here inject clicks into a game or work around any server limit. They measure your hand. What you do with the number is between you and the server rules.