System checks for Object for every test

So have you 'merged' your name map(s) into the other project(s)?  This may help.

https://support.smartbear.com/testcomplete/docs/testing-with/object-identification/name-mapping/how-to/merge.html

What is the exact error message you are receiving?

If you are new to TestComplete, I suggest you read and understand https://support.smartbear.com/testcomplete/docs/testing-with/object-identification/index.html 

When it fails, what is the error in the log? 

Do I understand your description correctly?
-Project A has NameMapping and Script using aliases to that NameMapping. In Project A, from the script you can perform item highlight and it works.
-In Project B, you add existing item of NameMapping and Script from Projact A, and from Project B script when you perform item highlight it fails. 

It would help to start with a few more details about the problem.

"What is the exact error message you are receiving?"

When it fails, what is the error in the log? 

• In Tools\Current settings, there is an object search strategy.  
• In Options\Engines there are name mapping settings.

Have you checked for issues with your NameMap?  I have seen issues like this.  In one case the path set to the NameMap in the project '.mds' was the problem.  The '.mds' file should not be manually modified but can be opened and examined as XML. 

In another case the NameMap was corrupt but I was able to recover by restoring an older NameMap from source control. 

The NameMap itself is stored in a NameMapping.tcNM file.  This is also an XML file that can be examined but I would not recommend modifying the content. 

In both cases this occurred in a situation similar to yours where two different projects were modifying the same NameMap.

If you are still stuck you may want to open an issue with the support team.

Just to rule out the simplest question...  "If I am trying to run tests, each and every test tries to identify the objects that are declared as aliases in each script"  This is normal behavior.  If an object is found in one test is it not stored for use in other tests or if it doesn't have scope in the code block you are in.  

Here is another point of interest in Options\Engines\Name Mapping...

Name Mapping Extended Search 

Notes

• Extended Find searches down the entire object hierarchy. The search is not depth-limited.
• Extended Find uses breadth-first search, regardless of the value specified in the Object search strategy project property (it affects the Find methods only).
• Searching for objects in large applications may take a long time. If you get warnings in the test log that a search takes too much time, try doing the following:
  • Add more objects to the object’s parent hierarchy in the Mapped Objects tree to narrow the search. For instructions, see About the 'Search for a Mapped Object Took Too Much Time' Warning.
  • Increase the project’s Auto-wait timeout.
  • Set the NameMapping.TimeOutWarning property to false. This will stop TestComplete from posting warning messages to the log if the search takes too long.