Wow, thank you all for your responses. I'll try to cover everything asked and commented above.
Keep in mind that I am quite new to this oil ordeal and I don't know much. Only what I found out reading this forum and looking at oil specs. I used to buy OEM oil for my cars but for this one I just cannot get it. Dealer said it's hard to get and it costs more than 30 USD per quart which is insane.
Nissan had 3 engines in 350Z's.:
2006 VQ35DE RevUp
The one with oil burning issue was the RevUp engine. Oil burning (supposedly) came from improper exhaust gas scavenging. Basicly engine was sucking in ceramic dust from the primary CATs scoring cylinder walls.
Let me clear things out with oil pressure. I take a car for a drive for about ten miles of warming up. I stop and check the PSI gauge. It reads 28-29 PSI with my previous oil (Nissan Fanfaro 5W-30) just before the service due. Used for 5.000 miles. When I did the oil change with Mobil1 5W-30, after this warm up drive and some stop and go warm city miles my PSI at idle was more 25-ish.
Bear in mind that this is immediately after oil change. So my old oil was holding pressure better.
I cannot find exact specs on 100ˇC viscosity on Nissan/Fanfaro, but it goes somewhere from between 11.5 - 12.5, for their Fanfaro ESX, LSX, EDX Full Synthetic oils. While Mobil1 ESP 5W-30 is 11.8. So viscosity is different.
@Overkill: The car has 22.000 original miles. It was bought from first owner. The car is in my possesion for 6.000 miles and I did one oil change. Oil consumption was 0.2 Quarts. I don't even drive my car in winter months. From the state of the engine and oil consumption I would exclude loose clearances and worn oil pump.
@Loneryder: Exactly, I used C3/C4. I had so many variables in my head I overlooked this part when getting my oil, which was a PITA due to this corona virus lockdown. C3 has lower SAPS and is more "clean" oil for the engine and CATs. I doubt this would make it up for different oil pressure. But oil will get replaced soon with A3/B3. I think @KameleON summed it up good. Regarding SAPS; C3/C4 oil cannot hurt gasoline engine, I suppose. The only issue that might affect the engine in the long run would be higher wear due to low SAPS (lover Phosphorus ans sulfur), if I am getting this right.
@Farnsworth: Honestly? Oil pressure is very close to OK. I will explain.
So, regarding my oil pressure issue. As VQ35DE RevUp had the oil consumption issue, VQ35HR has Oil Galley Gasket issue. This gaskets are used in an oil passage on the front of the engine block. Made out of paper. And as the car miles go beyond 30.000 more of these gaskets become prone to cracking and failing. When that happens oil pressure starts to drop. Usually it's not in an instance but through miles driven. If you don't get it fixed, pressure will eventually go to zero and you all know what follows.
(not my car)
There's an easy math for old V8 engine. They used to say 7PSI of pressure for every 1.000 rpm. I am not really sure if this still stands today for these new engines.
Nissan says there should be at least 15 PSI of pressure. That is the minimum. So I am still good. I am mentioning idling PSI because that is the only constant measurement I can get. And knowing, when I push the car and oil heats up, and idling pressure drops, there could potentially be risk with oil film not being sufficent enough at 7.000 RPM to keep bearings from damage.
And I hear you all on 0W-40 advice. And will probably go with it. My head just cannot get around the fact, why the Nissan Engineers didn't consider using 40 weight. So 30 should be sufficient?
Everyone is going with 5W-30 now. Even BMW M2 uses 5W-30 and it's a turbo engine. Cannot imagine how that oil keeps up with temps. Though BMW's tend to have bigger sumps than 5 quarts like 350Z and many other Japanese cars.