In Superbike & GP & other large scale competitions you do see the same sponsors supporting a multitude of riders. Whether they approach the riders team or wether its the other way around - i dont know.
In little ole New Zealand, - If a rider was wanting to approach the same sponsor & they were aware of the current riders support, I would say its lilekly better to approach the team & rider as one group on the basis that the you "might be" able to bring "the team" even more exposure for a cost that wont double the outlay (assuming that some costs travel & accomadation + bulk buying discounts actually decrease the overall outlay).
Other unkown issues (other than saving money) - need to be considered in this scenario though I.E rider / team personal compatibilty, + also the chosen machinery would need to be the same in suspension & tyre & brand if the goal was to actually develop both bikes/riders to increase the pace.

So my opionion is - that it is a viable option to approach a riders existing sponsor , however the method of the approach in our small market needs to be structured with some care so as not to damage the current riders "deal" & to genuinly benefit the sponsor as a whole.

In the scenario that its the sponsor that has approched a new rider in the first instance (instead of the other way round), then the gloves would be off.

Back-up option: Keep in mind If the "other rider" is on the bones of his / her arse financially, - then they may be just applying a shotgun approach to potential interested parties for support , & be sending there CV to a multitude of companies/people. ?