When service centre replace a motherboard is it new or refurbished?

Parts used for repairs may be new or remanufactured, at Apple's option. Apple guarantees that they meet the same high quality standards as new.

Some parts may not meet the Commerce standards for "new" e.g., may have been installed in a different machine at some point.

This is stated explicitly in their warranty and service agreements. You also relinquish any right to keep the old parts.

<<Does the parts used for repair meet the same high quality standards as new?

and do they function properly like a new part?>>

Yes, to both.

If you find the repair has not solved the problem, the repair itself has its own modest warranty. You can return your computer to the servicer and ask them to "make it right".

Most likely it will be a refurbished part. About the only time you may get a brand new part is getting a repair just after a new model is released.