Is it alright to ask the hiring manager and HR what is their hiring budget for a role?

I would say its ok, but I would phrase it as what kind of experience / pay grade are they looking for.

The background was in 2010 when I left a local company and was looking for another job. One of the recruitment firms asked me to attend an interview (which I was somewhat not keen on).

And it turns out the hiring manager's budget was about 20% lower than my expected pay, the location was in changi (i stay jurong), and I was expected to report to folks that might be less experienced than me. It was a horrible matchup.

So I do think its a fair question, regardless of what HR thinks. If HR insists on underpaying vs the market, then at least you know what you are getting yourself into.

Nowadays a lot of companies want to hire experienced, but pay up to 20+++% less (usually restructuring and adjusting the role to be one to two pay grades lower)

I've interned in HR previously and have had the opportunity to sit in interviews.
It might be a lil' sensitive too pose such a question if the underlying purpose is to use that as a negotiation for your salary. And furthermore, one must understand that within a role there are various grades as well hence the role isn't a determinant of your salary. Salaries should be negotiated based on your experience and what you can contribute to the company rather than what is budgeted.