[BUG]: Multiple users can register with the same display name causing matchmaking conflicts

[compiler041]

Bug Description

Problem

Multiple users can register or update their profile to use the same display name.
Since display name is used as the unique identifier in matchmaking, this causes
conflicts where two users with identical names cannot be distinguished in the
matchmaking pool, leaderboard, and debate rooms.

Steps to Reproduce

1. Register two accounts using emails like user1@example.com and user1@test.com
2. Both will receive the same default display name derived from the email prefix
3. Both users appear identically in the leaderboard and matchmaking pool

Expected Behavior

Each user should have a unique display name. Duplicate names should be rejected
at signup and when updating the profile.

Impact

• Leaderboard shows duplicate entries
• Matchmaking pool cannot distinguish between identically named users
• Debate room participant display is ambiguous

Environment

• Backend: Go + MongoDB
• Frontend: React + TypeScript

Steps to Reproduce

No response

Logs and Screenshots

No response

Environment Details

No response

Impact

Critical - Application is unusable

Code of Conduct

• I have joined the Discord server and will post updates there
• I have searched existing issues to avoid duplicates