The unwillingness or inability to share a budget for a product or a service hurts both the bidder and the government purchaser. It leads to worthless proposals which far exceed a budget and wastes the time of bidders whose products are services are outside the budget.
I'm not a contract specialist, so I don't know why the budget is often treated as a secret when in fact the budget was determined before a solicitation was published and is often available in public documents. In this case, why not provide a link to the budget document which has been published?
The idea that stating the budget will inflate prices because otherwise contractors would have been less without knowing the budget is a myth. If anything, discussing a budget leads to more realistic and successful purchasing.