Could I be allergic to my titanium implant?

Unlikely. Titanium is one of the metals least likely to cause allergy problems. Nickel or cobalt are the most likely metal allergens.
Unlikely. Unlikely related to titanium plate. Consider other causes.
Dental or ortho? Allergies to titanium orthopaedic implants are so rare that they are "reportable" - this means that that they can be written up and published in a scientific journal. There are however a number of reports of allergies to titanium dental implants.
Rare. But like anything, it is possible. Studies vary and continue to come out and percentage of people affected range from 0.6% in patients with implants to as high as 5% in patients that have a previously known metal allergy. If you have concerns, speak to your surgeon.