The conflict you are receiving is due to Budget Name must be unique.  Concatenate Budget and Budget Classification for unique Budget Name, then the data integrity issue will go away.  This can be done using Budget Name and Budget Classification Name or Budget Name and Budget Classification Code or some other uniqueness.

I already created Budget with unique names, but it gave me this error:

 Financial year "2022" is already used by budget "xyz"

Agent "Budgets Agent" returned error :'-2009'

Please review the index constraints using AM Designer application for budgets.  There you will see associations for budgets, budget centers, and Divisions must be unique.

Thank you for your clarification, but what if i want to create multiple budgets for same budget center and same division, but for different classifications, is this applicable?

As shown in below chart in user guide:

It is applicable and what you write is just like those pics show.  Budgets and Budget Classifications is a 1:1 link.

Hint:  "Budget 2001" and "Budget 2002" in the pic are child Budget Centers...not Budgets.

so as you described it is applicable, but what about index constrains for division and budget center, it prevents me to create multiple budgets for same division and budget center. 

As Shown in below screenshot

Your "multiple budgets" you write of IS NOT unique for same division and budget center.  The index constraint is at the Budget table level which means 1 Budget can only be associated with 1 Budget Center / Division combination.

So based on you clarification, if i have budget center "x" and division "2022-Annualy ", i can't create two budgets with following data:

Budget Name: x-2022-Annualy

Budget Center: x

division:2022-Annualy

Classification: Hardware

Budget Name: x-2022-Annualy

Budget Center: x

division:2022-Annualy

Classification: Software

You are not creating two budgets.  You have one budget with name "x-2022-Annualy".  Each budget name must be unique.