When I start to drive in the morning my steering is very tight and then loosens

Can I ask what makes you think its that over the power steering pump? The mechanic said it's hard to tell the difference?

40 years of the same issues. Usually after sitting overnight and hard to turn first thing, the gear is bad. You could change both if you want to have it all to be new.

You are really helpful! One more question, mechanic thinks probably power steering pump over rack because usually goes up first, car has 269,000 miles on it, I have heard that the power steering pump would make a noise if it's bad? There are no signs of leaking fluid? Any suggestions on that?

The pump is high mileage for sure. Have had many cars with your same symptoms of hard to turn initially then loosen up shortly after. The gear or rack n pinion change eliminated the issue. Flush the pump and lines good as well. The pump is old as well and it wouldn't hurt to consider that as well in the near future.