Do I need to get a new AFSP approval if I change my instructor or aircraft?
An AFSP application is valid for one provider, one aircraft category (TSA category, not FAA category), and one training event (single-engine, IR or multi-engine). If you change any of those things, you need to submit a new application.

If you're at a part 61 or 141 school that has multiple instructors, the approved provider is usually the school, not the individual instructor. So you can receive training from any instructor there without an issue.

If you're getting training directly from a freelance part 61 instructor, then it's possible that the instructor himself is the approved provider. In that case, you would need to submit a new application to get training from a different instructor or school.

In either case, your school or instructor should know exactly who's registered with the TSA as the provider.

As for the aircraft, you only need to submit a new request if you want to train in a different aircraft category. Initial training is usually done in category 3 aircraft (maximum MTOW of 12,500lbs) and you can train in one or many physical aircraft, as long as they're all in the same category. But remember that SEL and MEL are different training events so if your current approval is for SEL training you can't receive MEL training, even if the aircraft is under 12,500lbs.

Links to specific pages on the AFSP website don't work properly, but their FAQ is reasonably clear and you can email their helpdesk at AFSP.Help@dhs.gov if you need more information. AOPA has a useful guide, and the regulations (including category definitions) are in 49 CFR 1552.