All depends on what you have.
Is the new handle ( known as door furniture ) going to cover the existing paint marks etc as the existing " handle", saves repainting.
Is the new handle the same backset as the existing, backset is the distance from the front edge of the door to the centre of the handle,the domestic industry standard is 60 mm .
Some privacy latches do not match others,so some mods may need to be done.
Some door furniture is mounted on the door through a standard 54 mm hole,others use any number of mounting configurations.
Best take one off and take it to a hardware shop and ask for options that don't involve repainting or hopefully drilling larger holes through a door that already has holes in it as trying to drill over existing holes can't end in .....swearing.[
Mortise latch
Backset, 60 mm from edge of door to centre of handle,knob
.
Striker plate.
Mortise latches are generic your existing should suit the new handle so no worries there.
The same if using a passage set or privacy set,the locking mechanism in internal doors is in the handle/ knob nothing to do with the striker plate.
Striker plates are generic as well.
Cost is a factor and the name says a lot in that department,Lockwood etc.
Internal door furniture is pretty much same o same,Bunnings etc have a large range and there are many that look the same,have a different name and yet do job.
Gainsborough is one that Bunnings sell but under another name,try to find one the same colour/ finish as the existing, as the shape of the striker plates vary so buying one the same colour won't need the striker plate changed.
The pics I have posted are of Gainsborough handles,they require 3 holes for mounting which consists of 4 threaded screws into a ferrule in the centre and the mongrels are for ever moving,the copped a bit as the kids were growing and couldn't quite reach so they pulled on them as kids do ,therefore putting strain on the mounts.
Hope this helps.