feat(browser): Set page context instead of using request interface

[lforst]

Ref #15845

[github-actions]

size-limit report 📦

Path	Size	% Change	Change
@sentry/browser	23.25 KB	-0.08%	-18 B 🔽
@sentry/browser - with treeshaking flags	23.08 KB	-0.06%	-12 B 🔽
@sentry/browser (incl. Tracing)	36.67 KB	-0.04%	-13 B 🔽
@sentry/browser (incl. Tracing, Replay)	73.84 KB	-0.02%	-11 B 🔽
@sentry/browser (incl. Tracing, Replay) - with treeshaking flags	67.2 KB	-0.03%	-16 B 🔽
@sentry/browser (incl. Tracing, Replay with Canvas)	78.5 KB	-0.02%	-10 B 🔽
@sentry/browser (incl. Tracing, Replay, Feedback)	91.06 KB	-0.02%	-12 B 🔽
@sentry/browser (incl. Feedback)	40.4 KB	-0.03%	-12 B 🔽
@sentry/browser (incl. sendFeedback)	27.88 KB	-0.08%	-20 B 🔽
@sentry/browser (incl. FeedbackAsync)	32.69 KB	-0.03%	-9 B 🔽
@sentry/react	25.06 KB	-0.05%	-12 B 🔽
@sentry/react (incl. Tracing)	38.58 KB	-0.05%	-17 B 🔽
@sentry/vue	27.47 KB	-0.07%	-18 B 🔽
@sentry/vue (incl. Tracing)	38.36 KB	-0.08%	-28 B 🔽
@sentry/svelte	23.29 KB	-0.08%	-19 B 🔽
CDN Bundle	24.51 KB	-0.02%	-3 B 🔽
CDN Bundle (incl. Tracing)	36.7 KB	+0.01%	+3 B 🔺
CDN Bundle (incl. Tracing, Replay)	71.71 KB	+0.01%	+7 B 🔺
CDN Bundle (incl. Tracing, Replay, Feedback)	76.89 KB	+0.01%	+2 B 🔺
CDN Bundle - uncompressed	71.45 KB	+0.03%	+15 B 🔺
CDN Bundle (incl. Tracing) - uncompressed	108.64 KB	+0.02%	+15 B 🔺
CDN Bundle (incl. Tracing, Replay) - uncompressed	219.93 KB	+0.01%	+15 B 🔺
CDN Bundle (incl. Tracing, Replay, Feedback) - uncompressed	232.49 KB	+0.01%	+15 B 🔺
@sentry/nextjs (client)	39.87 KB	-0.05%	-19 B 🔽
@sentry/sveltekit (client)	37.08 KB	-0.06%	-21 B 🔽
@sentry/node	142.93 KB	-	-
@sentry/node - without tracing	96.13 KB	-0.01%	-1 B 🔽
@sentry/aws-serverless	120.48 KB	-0.01%	-1 B 🔽

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