test(browser): Add integration tests for setContext

[onurtemizkan]

Part of: #4333

[github-actions]

size-limit report

Path	Base Size (1c3f85a)	Current Size	Change
@sentry/browser - CDN Bundle (gzipped)	20.31 KB	20.31 KB	+0.02% 🔺
@sentry/browser - CDN Bundle (minified)	64.7 KB	64.7 KB	+0.01% 🔺
@sentry/browser - Webpack	22.29 KB	22.29 KB	0%
@sentry/browser - Webpack - gzip = false	76.23 KB	76.23 KB	0%
@sentry/react - Webpack	22.33 KB	22.33 KB	0%
@sentry/nextjs Client - Webpack	46.45 KB	46.45 KB	0%
@sentry/browser + @sentry/tracing - CDN Bundle (gzipped)	28.45 KB	28.45 KB	+0.02% 🔺