ref(core): Keep client-logger map on carrier & avoid side-effect

[mydea]

Instead of having a side effect of writing on the GLOBAL_OBJ, we can use a global singleton instead.

Part of #16846

[github-actions]

size-limit report 📦

Path	Size	% Change	Change
@sentry/browser	23.96 kB	-0.02%	-3 B 🔽
@sentry/browser - with treeshaking flags	22.43 kB	+0.03%	+6 B 🔺
@sentry/browser (incl. Tracing)	39.83 kB	+0.01%	+1 B 🔺
@sentry/browser (incl. Tracing, Replay)	77.95 kB	-0.02%	-11 B 🔽
@sentry/browser (incl. Tracing, Replay) - with treeshaking flags	67.77 kB	-0.01%	-3 B 🔽
@sentry/browser (incl. Tracing, Replay with Canvas)	82.64 kB	-0.02%	-12 B 🔽
@sentry/browser (incl. Tracing, Replay, Feedback)	94.75 kB	-0.02%	-12 B 🔽
@sentry/browser (incl. Feedback)	40.66 kB	-0.02%	-5 B 🔽
@sentry/browser (incl. sendFeedback)	28.64 kB	-	-
@sentry/browser (incl. FeedbackAsync)	33.54 kB	-0.01%	-2 B 🔽
@sentry/react	25.68 kB	+0.02%	+4 B 🔺
@sentry/react (incl. Tracing)	41.79 kB	-0.01%	-1 B 🔽
@sentry/vue	28.39 kB	+0.01%	+2 B 🔺
@sentry/vue (incl. Tracing)	41.62 kB	+0.04%	+13 B 🔺
@sentry/svelte	23.99 kB	+0.04%	+8 B 🔺
CDN Bundle	25.26 kB	+0.11%	+27 B 🔺
CDN Bundle (incl. Tracing)	39.52 kB	+0.05%	+16 B 🔺
CDN Bundle (incl. Tracing, Replay)	75.51 kB	+0.03%	+18 B 🔺
CDN Bundle (incl. Tracing, Replay, Feedback)	80.95 kB	+0.01%	+3 B 🔺
CDN Bundle - uncompressed	73.72 kB	+0.06%	+44 B 🔺
CDN Bundle (incl. Tracing) - uncompressed	117.11 kB	+0.04%	+44 B 🔺
CDN Bundle (incl. Tracing, Replay) - uncompressed	231.29 kB	+0.02%	+44 B 🔺
CDN Bundle (incl. Tracing, Replay, Feedback) - uncompressed	244.1 kB	+0.02%	+44 B 🔺
@sentry/nextjs (client)	43.41 kB	+0.03%	+11 B 🔺
@sentry/sveltekit (client)	40.29 kB	+0.01%	+1 B 🔺
@sentry/node	167.64 kB	+0.01%	+2 B 🔺
@sentry/node - without tracing	100.37 kB	+0.01%	+10 B 🔺
@sentry/aws-serverless	128.49 kB	+0.01%	+2 B 🔺

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