Is it safe to use a 125PSI rated auxiliary air tank with a 150PSI air compressor?

If you have the Porter Cable compressor pictured, there's a regulator on the outlet. You could set that to 100 psi and fill your tank and be fine. As everyone has mentioned if you fill it to 150 psi you'll probably kill yourself or at least soil some underwear when one of the end caps explode.

Don't do it!!

Just buy a pressure regulator to put between the two tanks that will decrease the pressure to below 125. You can get these for $17 at home depot (https://www.homedepot.com/p/Husky-1-4-in-NPT-Regulator-with-Gauge-41155HOM/205331831) and maybe even cheaper elsewhere.

I'm terrified of the tank exploding and covering me with shrapnel.

This is a valid concern. You should treat pressure vessels as if they were bombs, because in some sense they are. Failures are "Not Good".

I have a hard time believing a mere 25PSI or 10PSI (depending on whether you believe the manual or the relief value) would cause the tank to explode.

Well, in some sense, you are right. If you take a tank rated to 125 psi and put 135 psi in it, it probably won't have an issue. But the magic word there is "probably". Let's quantify that. We can't answer the question "is it safe?" as if it was a yes or no answer. It's more like a probability that there will be in issue.

why should we think in terms of probability? Well, what determines the strength of the tank? For starters, there is the thickness of the wall. There are always manufacturing variations. Some tanks will be a tiny (thousandths of an inch) thicker, and others thinner. So due to this, some will fail at a slightly higher pressure, and others at slightly lower pressure. There are many other sources of variation, but this easiest to think about, so let's stop here.

Now I'm going to make up some numbers here, but it will illustrate the point. At the rate pressure, 125 psi, the probability of failure is very very small. Maybe not zero, but small. Let's just say 1 in 10 billion for sake of argument (i.e. every person in the world has an air tank, one of them explodes at the rated pressure). Now, if you go a little above the rated pressure, say 130 psi, now maybe 10 out of those 10 billion have an issue. It's still a pretty small number of tanks that have an issue, so you keep going. At 135 psi, now maybe 1000 out of the 10 billion (or 1 out of every 10 million) have an issue. At 140 psi, it's 1 out of every 100,000. At 145 psi, it's one out of every 10,000. and at 150 psi, it's 1 out of every 1,000.

Again, I'm just making up the numbers, but you get the point. It's like a game of russian roulette. If you are just a tiny bit over in pressure, you're "probably" good. But the higher you go, the riskier it becomes. Most tanks won't have an issue at a little bit over the rated pressure, but how do you know if yours is the unlucky weakest one?

Given that you can stay within the rated pressure with a $17 part, it's just not worth taking the risk

A 10 or 25 psig overpressure of a tank is well into a potential failure for the tank. The 140 psig safety valve is used because that accounts for the true pressure capability of the tank plus the tolerance in lift pressure for the safety valve. If you replace it with a 150 psig valve, you are risking the tank building up pressure past it's breaking point to reach the point when the 150 psig safety valve opens - the tank will vent via rupture before the valve has a chance to do anything.