So long as you remain within the approved aircraft specifications, to include any engine or prop changes, there is no required additional "authoritative certification" needed outside of an A&P certificate.
I am aware that FAA A&P privileges are a little different from those I'm familiar with in Canada, as are the FAA standards for previous certification. An (unrelated to the topic) example is that I understand that a previously certified airplane may be moved into an "experimental" category at the owner's option. This is not permitted in Canada, unless there is a genuine experimental flight test program to be carried out on the airplane, for which an accepted flight test plan will be followed. This difference was an impetus for the creation of the Canadian Owner Maintenance category, to enable some airplanes to be voluntarily moved from certified to a status part way toward "amateur built" (which a previously certified airplane is not).
On the main topic, very dependent upon what certification or other records accompanied an airplane part, and what nation the work is being certified in, a licensed aircraft mechanic might be allowed to accept it for installation - if it "conforms to the type design" (for which the records probably help) and is "fit and safe of operation". I expect that if the documentation which accompanies a part is considered incomplete, it'll take more time and consideration by the mechanic to accept it for installation. For a couple of parts to repair an airplane maybe it's worth it. A major number of the parts to construct a whole plane - probably not worth the time it would take for the mechanic to be at peace with the determination. A factor in this is the confirmation of the "fit and safe for installation" part of the requirement, as donor parts from wherever will require detailed inspection for conduit and geometry.
Even an STC approval is not required since per the Part 1 definition of a major alteration states any alteration listed in an aircraft specification is not considered a major alteration. And there have been a number of engine/prop changes listed in such a manner.
Correct, if the proposed engine and/or prop is listed on the TCDS, or STC approved. But, if neither approval is available, an STC will be required for the airplane to be fully certified following the changed engine/propeller installation. I am aware of a few situations where the installation of a different engine and or propeller created an unexpected unsafe condition - I've found a few during flight testing. This is a particularly important consideration if the airplane also has other modifications. "Multi mod" airplanes are receiving more scrutiny these days to assure that they meet the requirement that: "Compatibility of this design change with previously approved modifications must be determined by the installer". Such determination could be an aircraft certification type activity, and may be beyond the scope of some installers.
Example of the situation:
Cessna A185F being rebuilt following groundloop damage - wings replaced. The licensed mechanic signed out the repair, understanding (but not mentioning in the work report) that Cessna 182Q wings were installed in place of the wings intended for an A185F. The 182 wings, with some modification, will fit and work - but are not approved, and they did not do all of the required modifications. But, the airplane was returned to service, flew many years, and passed several annual inspections without this non compliance being noticed, nor the several unsafe conditions being noticed or resolved - until a prospective purchaser did a good pre purchase technical record review, and found this major discrepancy. Questions were asked, unsafe conditions suspected, and the airplane grounded. Fuel and control systems differ between the types, and a major difference, the stall warning system. Story of making it right here:
Modifying a Cessna stall warning system
In this situation, a $450,000 airplane was rebuilt with errors related to the use of wrong or undocumented parts, and in a number of aspects, was unsafe. The owner pilot, and several licensed mechanics did not notice these unsafe conditions for years. It cost the value of a decent smaller Cessna to make it all right.