Okay, I just checked my 6.3V 1800 uF Panasonic FL caps (sorry for the delay).
The top cap in this picture of yours definitely looks genuine: https://www.badcaps.net/forum/attach...1&d=1459118731
Basically, the "[M]" logo is flush with the "+105°C FL" text, just like on mine and is the right size. I also compared it to several 16V 1500 uF FL caps (10x20 size), and again everything looks the same.
That said, the bottom cap could still be genuine, despite the "[M]" logo not being flush with the text. Manufacturing differences, perhaps? All I can tell you is that I also have some Panasonic FJS 6.3V 820 uF caps, and they had their "[M]" logos not flush with the text above them, just like the bottom cap in your picture. So again, there is a chance that it is genuine, too.
Speaking of genuine caps, all of my Panasonic caps are board pulls from ASUS or Xbox 360, so it's unlikely they are fakes.
Do you know if capacitors have a certain shelf life? If so what's the best way to store them so they last longer?
They do, but that can vary a lot, depending on the capacitor type (ultra-low ESR caps are a bit more unstable than "entry" and "mid" -level low ESR).
In general, 2-3 years in storage is okay. After 5 years, they may start to degrade a little (but probably not enough to matter at all). After 5 years, though, it's recommended you reform them. See this for more details: https://www.badcaps.net/forum/showthread.php?t=50530
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