No. For one, the Shyamantaka Mani is described in the Puranas as having many spectacular and obvious supernatural properties. But even if you take this as extreme poetic exaggeration rather than literally (or think that these properties have somehow become dormant or hidden), there is a glaringly apparent problem with this theory: the Shyamantaka Mani wasn’t a diamond at all. It was a ruby.
The Kohinoor diamond was indeed an old and treasured Indian diamond plundered in war by the British invaders and still grossly held by the British royalty to this day, an unrepentant testament to the rapacious cruelty of the colonial era. But it is not the Shyamantaka Mani.