ref(browser): Improve type safety of breadcrumbs integration

[mydea]

In the process of looking into #7373, I noticed that the current breadcrumbs implementation uses a lot of any types, making it harder to refactor stuff and ensure we do not regress.

This PR updates this to use proper types, both in instrument site and in breadcrumb site.

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser - ES5 CDN Bundle (gzipped + minified)	20.14 KB (+0.1% 🔺)
@sentry/browser - ES5 CDN Bundle (minified)	62.53 KB (+0.03% 🔺)
@sentry/browser - ES6 CDN Bundle (gzipped + minified)	18.77 KB (+0.1% 🔺)
@sentry/browser - ES6 CDN Bundle (minified)	55.53 KB (+0.07% 🔺)
@sentry/browser - Webpack (gzipped + minified)	20.51 KB (+0.08% 🔺)
@sentry/browser - Webpack (minified)	67.03 KB (+0.06% 🔺)
@sentry/react - Webpack (gzipped + minified)	20.54 KB (+0.08% 🔺)
@sentry/nextjs Client - Webpack (gzipped + minified)	48.28 KB (+0.02% 🔺)
@sentry/browser + @sentry/tracing - ES5 CDN Bundle (gzipped + minified)	27.27 KB (+0.07% 🔺)
@sentry/browser + @sentry/tracing - ES6 CDN Bundle (gzipped + minified)	25.54 KB (+0.09% 🔺)
@sentry/replay ES6 CDN Bundle (gzipped + minified)	43.11 KB (0%)
@sentry/replay - Webpack (gzipped + minified)	37.16 KB (0%)
@sentry/browser + @sentry/tracing + @sentry/replay - ES6 CDN Bundle (gzipped + minified)	60.92 KB (+0.04% 🔺)
@sentry/browser + @sentry/replay - ES6 CDN Bundle (gzipped + minified)	54.25 KB (+0.03% 🔺)