Client Version: Fabric 1.20.4
Server Version: 1.20.4 Velocity

Nicknames? Yes
Format: <prefix> <nickname>: <message>
Output after adding my own nickname:

I assume you're using 0.10.32 and this only happens with nicknames, right?

What does it show in the tab menu / player list?
If you click a player's name, do you get a /tell command to DM that player?

If the nicknames don't match the tab menu name or the /tell command doesn't match the profile name, there's no way Chat Heads can associate the message with its sender.

What you can try is to add a name alias in the config:
Etanarvazac -> your_profile_name

The server has custom click and hover events (e.g.: clicking on my rank auto inputs the /ranks command) on player name. So that wouldn't work as it is. I made a dummy server and it doesn't even work on there. Yes, I am using 0.10.32

Edit 1: It doesn't even work on local worlds both offline and open to LAN:

Edit 2: I even manually put it in the config:

{
	"offsetNonPlayerText": true,
	"senderDetection": "UUID_AND_HEURISTIC",
	"smartHeuristics": false,
	"handleSystemMessages": true,
	"nameAliases": {
		"Mod | Etanarvazac": "Etanarvazac"
	}
}

Just for it to be ignored:

Edit 3: Joined a random server and again, it didn't work, and with setting my nickname to match:

Also, if the chat messages don't that clickable /tell command, that is an issue that should be reported to the used chat plugin.

I made a dummy server and it doesn't even work on there.

Is that dummy server modded or not?

It doesn't even work on local worlds both offline and open to LAN:

Wait, it doesn't even work in singleplayer?
Then this is a mod conflict with some mod you've installed.

Please send your mod list so I can have a look.
I hope you're not using OptiFabric, since OptiFine is known to break this mod since around 1.20.1.

I hope you're not using OptiFabric, since OptiFine is known to break this mod since around 1.20.1.

Looking at this again, the font your chat uses looks suspiciously like a HD font.
Here's the obligatory tip to use Sodium and co instead.

Since there's no further info, I'm gonna close this issue.