feat(telemetry): Add process tags to instrumentation telemetry

[wantsui]

What does this PR do?

Adds process tags to instrumentation telemetry for https://datadoghq.atlassian.net/browse/AIDM-255

Motivation:

https://datadoghq.atlassian.net/browse/AIDM-255

Follows the work done in #5033

The Python implementation that I followed can be found here: DataDog/dd-trace-py#15227
Change log entry

None.

Additional Notes:

How to test the change?

docker compose run --rm tracer-3.3 /bin/bash
bundle exec rspec spec/datadog/core/telemetry/integration/telemetry_spec.rb

[github-actions]

Typing analysis

Note: Ignored files are excluded from the next sections.

Untyped methods

This PR introduces 1 partially typed method. It decreases the percentage of typed methods from 55.4% to 55.37% (-0.03%).

Partially typed methods (+1-0)

❌ Introduced:

sig/datadog/core/telemetry/request.rbs:11
└── def self.tag_process_tags!: (Hash[Symbol, untyped] app, Core::Configuration::Settings config) -> void

If you believe a method or an attribute is rightfully untyped or partially typed, you can add # untyped:accept to the end of the line to remove it from the stats.

[pr-commenter]

Benchmarks

Benchmark execution time: 2025-12-10 21:58:59

Comparing candidate commit 0ace8cd in PR branch add-process-tags-to-telemetry with baseline commit 80b8b5a in branch master.

Found 0 performance improvements and 0 performance regressions! Performance is the same for 44 metrics, 2 unstable metrics.

[datadog-datadog-prod-us1]

✅ Tests

🎉 All green!

❄️ No new flaky tests detected
🧪 All tests passed

🎯 Code Coverage
• Patch Coverage: 97.37%
• Total Coverage: 95.24% (-0.00%)

View detailed report

<sub>This comment will be updated automatically if new data arrives.
* Fix with Cursor requires Datadog plugin ≥v2.17.0
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