Implement consistent logging across the application

[conico974]

Introduce a centralized logging mechanism to ensure uniform logging practices throughout the app. Move the logger initialization to a dedicated file and utilize the logger in various components for improved traceability and debugging.

For it to work perfectly in OpenNext we need this opennextjs/opennextjs-aws#914.
Right now the requestId from the middleware will be different from the one in the server function, and no header will be set

[linear]

RND-7359 Create a custom logger for GBO

[changeset-bot]

⚠️ No Changeset found

Latest commit: 63a0f28

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[github-actions]

Summary of the deployments:

Version 1

Version	URL	Status
Latest commit	https://9f247195.gitbook-open.pages.dev	✅
PR	https://pr3368.gitbook-open.pages.dev	✅

Version 2

Version	URL	Status
Vercel	https://gitbook-v2-nj19um8am-gitbook.vercel.app	✅
Cloudflare	https://ac08f4e2-gitbook-open-v2-preview.gitbook.workers.dev	✅

Test content

Site	v1	2v	2c
GitBook	https://9f247195.gitbook-open.pages.dev/gitbook.com/docs	https://gitbook-v2-nj19um8am-gitbook.vercel.app/url/gitbook.com/docs	https://ac08f4e2-gitbook-open-v2-preview.gitbook.workers.dev/url/gitbook.com/docs
E2E	https://9f247195.gitbook-open.pages.dev/gitbook.gitbook.io/test-gitbook-open	https://gitbook-v2-nj19um8am-gitbook.vercel.app/url/gitbook.gitbook.io/test-gitbook-open	https://ac08f4e2-gitbook-open-v2-preview.gitbook.workers.dev/url/gitbook.gitbook.io/test-gitbook-open

[argos-ci]

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Build	Status	Details	Updated (UTC)
customers-v1 (Inspect)	✅ No changes detected	-	Jun 23, 2025, 4:10 PM
customers-v2 (Inspect)	✅ No changes detected	-	Jun 23, 2025, 4:13 PM
default (Inspect)	⚠️ Changes detected (Review)	1 changed	Jun 23, 2025, 4:13 PM
v2-cloudflare (Inspect)	⚠️ Changes detected (Review)	1 changed	Jun 23, 2025, 4:15 PM
v2-vercel (Inspect)	⚠️ Changes detected (Review)	1 changed	Jun 23, 2025, 4:15 PM