I'm looking to renew one of the tyre on my MTB as it's on a dying stage. I see some tyres those have 40 PSI tyre pressure level and some have 60 PSI. They are both 26x2.00 so it can't be about width. Why would one manufacturer suggest much lower tyre pressure than the other? Does that show they have bad quality tyres?
No, the relationship you suggest is not the case. The opposite is almost true, but not in all cases.
The use cases where a 2" or 50mm wide tire should be ridden at 60 psi exist but are relatively uncommon. An example of such an application would be a cargo e-bike, tandem, or other very heavily loaded bike, or the disciplines that involve extreme drops. But in a lot of other cases where someone fills a 2" tire to that kind of pressure just to ride around casually, they're making the ride unnecessarily harsh and bumpy for no benefit.
Limits to pressure suggest a lightweight and supple casing, which is desirable as long as it's balanced well with durability for the application. High pressure limits suggest a very strong or even overbuilt casing, which again has applications where it's desirable, but not usually in performance-oriented riding other than the gravity disciplines.
So, in other words, there are some good 26x2.0 tires that say max 60 or 65, such as a trekking/cargo/touring/tandem/commute/ebike tire where the emphasis is on toughness and therefore the casing can handle that pressure whether it's useful or not. But meanwhile, many bad or mediocre tires have high nominal max pressure ratings simply because they're cheap and heavy.
The other thing about the relationship between quality, strength, and nominal max pressure is manufacturers have choices to make about how conservative to be in the number they make up. When casings do bulge or warp from pressure it's usually not all at once, but happens over time and is a result of load, pressure, rim width, environmental conditions, what kind of minor damage they're picking up, etc. The question then arises of whether they turn to rigorous testing, the lawyers, or the dartboard to actually come up with the number. All are possible but since we're talking about the bike business, it's probably usually lawyers and dartboard competing for first place and testing third, in order of likelihood or share of influence. But that said, lighter tires are more apt to show a max pressure that is closer to a real hard limit and possibly even pushing it, because light bike parts do their thing by flying closer to the sun.
Don't take the pressure specifications on tyres as anything more than an “idiot guide”: if you pump it to that value then you can be pretty sure it will neither spontaneously explode nor have unduly high risk of pinch flats – but that's about it. In most cases, the numbers will be far from ideal for whatever kind of riding you intend to do with it.
Typical pressures that make sense for MTB riding are between 1.5 and 2 bar / 20-30 psi. It's totally possible to go outside of that range, but you should only do it if you have a good reason for it. I variously run pressures between 1 bar and 4 bar in the same MTB tyres. For example, in mud and snow very low pressure makes sense.
This applies somewhat orthogonally of inherent tyre quality / reliability. With different tyres there may be different reasons for choosing particular pressure values. For example, light XC-oriented tyres are easily damaged. If you use them in gnarlier terrain than they were designed for, or with a heavily loaded bike, pumping to higher-than-ideal pressures helps somewhat protecting them from pinch flats. For downhill tyres this wouldn't be as much of a concern, on the other hand those have the problem of very high rolling resistance on tarmac or hard-pack, which can also be somewhat ameliorated by high pressure. In both cases, you sacrifice grip and riding comfort, but it may be worth it. And at least if using inner tubes, you can usually expect that a good-quality tyre will also cope with a lot more pressure than the manufacturer voiches for.