By vigorously stirring water with a glass rod, you could potentially increase its kinetic energy which would in turn increase the temperature of the water. Similarly, would increasing the energy of the bottle of water by dropping a bottle of water from 100m to 0m increase the temperature of the water? You would be increasing the kinetic energy of the water bottle system while falling, and water upon crashing into the ground. I've researched this concept before and found out that coherent kinetic motion has no effect on the temperature of water.
Vigorous stirring is exactly what James Prescott Joule did in his experiments, so the answer is "yes", the temperature goes up. Now the subtlety that you have corrently noticed is that coherent kinetic energy of motion for the fluid is not the same as a temperature increase. The resolution is that viscosity eventually damps out the coherent motion and turns it into incoherent molecular motion, i.e heat. Viscous dissipation is present whenever parts of the fluid have different velocities, and this exactly what happens when fluid is stirred.