Brightening an underexposed RAW later is usually not equivalent to getting the exposure right in-camera.

The main reason is signal-to-noise ratio: if you record less light, the image data starts noisier. Raising exposure afterward boosts both signal and noise, especially in the shadows, which can reveal color noise or casts like purple tint.

Also, the camera’s higher ISO setting is not always the same as a simple 2× digital multiply. On some cameras, even “extended” ISO may involve different internal handling, and RAW converters can apply ISO-dependent processing. Lightroom may treat an ISO 12800 file differently from an ISO 6400 file pushed +1 EV, particularly for noise reduction and color handling.

So the difference you see can come from both:

1. less light captured in the underexposed frame, and
2. different processing between in-camera/high-ISO RAW metadata and Lightroom’s manual exposure push.

If you want the closest comparison, try the camera maker’s RAW converter as well. In practice, exposing properly in-camera usually gives better quality than intentionally underexposing and pushing later, unless you are protecting highlights.