Hi,
> Is there not any way to map object once [...]
This is the purpose of NameMApping/Aliasing - to map once and reliably use the mapped object for all browsers and until the UI layout of the tested application is completely redesigned.
The problem here is that it is only you and your team (designers and developers) who know the UI layout of your tested application and what properties of web elements can be used for stable and reliable mappings.
TestComplete tries its best but there is no universal way here and good mapping always requires it to be done in advance, before recording.
https://support.smartbear.com/screencasts/testcomplete/reliable-tests-for-dynamic-objects/ from https://support.smartbear.com/screencasts/testcomplete/ page might appear to be useful.
