Why did one of my UPS batteries swell up but the other one looks fine?

Yes, a puffed battery is done, you had that right. The swelling is gas building up inside the cells, usually from the plates sulfating and breaking down after years of sitting at float charge. Never put a swollen one back in service and don't try to recharge it outside the case either. Take it to a battery recycler or a place that sells car batteries, most take them free. As for why one died and the other didn't, it's mostly luck of the draw. Two batteries in the same series string never age at exactly the same rate. Tiny differences from the factory get amplified over the years, and the weaker one ends up doing more of the work during every little sag and outage, so it wears faster and gasses while its neighbor still looks fine. It's the same reason you always see one dead cell take out a whole car battery. Could the UPS charger be at fault? possable but unlikely at this age. These little units just push a fixed float voltage through both batteries in series, and 5 or 6 years is honestly a full life for that kind of battery anyway. If your replacements swell up in a year or two, then I'd suspect the charger. If they last another 4 or 5, it was just age. On sizing, you don't need the exact rating. A 7.2 or 7.5Ah in the same case size is a drop-in upgrade and gives you a few extra seconds of runtime. Just check the terminal type, some come with the wider F2 terminals instead of the narrow ones, though most in this size use the same small spade. What you can't do is put a smaller one in, or mismatch capacities between the two. And yes, replace both as a pair. It sounds like an upsell but it's not. The old battery has lost capacity and has a diffrent internal resistance, so pairing it with a fresh one just drags the new one down to its level and kills it early. A matched pair costs a bit more up front and lasts far longer. Since your runtime is only 10 seconds of beeping you could even go a size up physically if there's room in the case, but the same-footprint 7.5s are the easy answer.

On my unit, I replaced the batteries last year and the beeping definately stopped. But I've seen UPS units starting to fail even when the batteries are fresh. If the unit is still kicking out after a while, the problem is likely the internal electronics, specifically the inverter itself.

I would check if the UPS is rated for the actual load you are running, like the fridge. Sometimes the internal components just can't handle the consistent draw, no matter how good the battery is. It's worth seeing if the UPS has a specific replacement module or if you just need to upgrade the whole box to handle the power demands better.

I agree with the diagnosis that a swollen unit is completely junk. I ran into the exact same issue last fall with a smaller backup unit and decided to replace the set even though one seemed to have just dropped off the charts. Always good practice to check the voltage on the failed terminals first, just to make sure the issue isn't just a loose connection rather than the whole battery pack failing.