I got this question after watching some cutscenes from Lego Star Wars: The Skywalkers Saga
Anakin just killed Dooku, and Grievous walked in, (unlike the movie) and the duo of Kenobi & Skywalker assume a fighting pose, with Anakin holding Dooku's saber.
Why didn't this happen in the original movie? Not the Grievous part but Anakin keeping the lightsaber to fight Grievous later, I mean.
Anakin dropped it in shock at the enormity of his actions.
Now he stood over a headless corpse that he couldn’t bear to see but he couldn’t make himself look away, and he knew it hadn’t been a dream at all, that he’d really done this, the blades were still in his hands and the ocean of wrong he’d dived into had closed over his head.
And he was drowning.
The dead man’s lightsaber tumbled from his loosening fingers. “I-I couldn’t stop myself…”
And before the words left his lips he heard how hollow and obvious was the lie.
History doesn't record why he didn't pick it up, but I would assume he simply wasn't inclined to do so, noting that he's mainly trained to use one lightsaber, not two. Having a second one is dead weight.