I've been searching around online for Genuine PS1 and PS2 memory cards because I lost my memcards recently, until I found fakes of the same memory cards. Do they work well? Also I've seen 128Mb PS2 memory cards, are they even genuine?
The only official PS1 memory card is 1 megabit (16 blocks of which 1 not advertised and reserved for the filesystem) - they only say "Memory card" on top (most accurate clones, which also are fine, say "1 Mega"), and other sizes are not possible - any larger card is multi-partitioned (and/)or an incompetent design
The only official PS2 memory cards are 8 MB or 16 MB (the latter wasn't made by Sony but officially licensed) but, while the specification does allow for larger sizes, the official name of a PS2 card was for many years "Memory Card (8MB) (for PlayStation 2)" - some developers took it too literally so even the official 16MBs don't have 100% compatibility
An original card or at least a good clone will implement MagicGate correctly - while there's no trivial test for that on a PS2 unlike on a PSP, you can try installing and running an update signed to the card's serial number (like FMCB, more or less official DVD Player updates, or updates for a few games - I think some of those are in the Socom series)
As far as I know from reading years ago, that wasn't something that happened "on accident". Sony intentionally forced certain companies, usually big ones such as Square/Enix, Capcom and Konami to make checks on the memory cards to see if they were from Sony. If not, they'd refuse to save or create fake, corrupted save files. This was most easy to find with 16MB and above cards.
God Hand from Capcom and one of the Silent Scope games from Konami are two games I know first hand, but I know there are more, and also happened with PS1 games. It's... Very nice from Sony, but what do you expect from a company that doesn't even let you format your memory cards without a third party involved or freaking homebrew...
As of today I have one original PS1 and PS2 MC, and also two third party ones for each too. I've yet to find any issue with the third party PS1 one, it actually works pretty damn well, looks exactly like the one in alexander1970's photo and I bought it from aliexpress, which is not your "100% quality" online store at all , but I'm sure there's quite a bit of a lottery with these aftermarket things.
What I'd recommend you is to have a way of launching FreeMCBoot and make backups of your save files from time to time. You can transfer PS1 saves from a PS1 MC to a PS2 MC through the systems native menu, and from there to a USB stick through ULauncher. That's what I've been doing since I got a PS2 again, both to preserve my saves, and because of the horrid sizes of the memory cards not letting you save many games at once, specially on PS1, 15 slots are not enough one quite a few games can take more than one.
I don't recommend getting a third party PS2 MC that's not 8MB just to be safe. That extra space sounds great but the way most of them achieved it wasn't safe at all, using things such as compression that would lead to corruption of saves.
I imagine you also know that with "128Mb" they mean Megabits, so in reality it's 16 Megabytes, which mind you, if they worked, 16MB are quite enough (unless you play sport games, those can take huge amounts of space, some even half of a 8MB card).
If you try to buy official cards, pray that they've not seen too much use, because wasting time to find one only to be worn out and failing sucks tremendously.
All this is so much less complicated on the GameCube... I've got two third party generic 128Mb cards for who knows how many years and they still work perfectly, and on GC the size of the card doesn't cause any issue whatsoever, I have dozens upon dozens of saves on each and still lots of free space.
In order to contribute to this subject I will share a picture of one of my PS1 cards, it seems to be a clone (I thought it was original). The point is, I have 2 of these for like a year know, and I have NEVER had a fail save, or lost any saved games, not even once.
I guess they are pretty good clones? Because I doubt they are "cheap" originals, right? If anyone can confirm, please do.
I'd recommend eBay. There are tons of Sony branded memory cards and they can be had for around $10.
I had so many game saves corrupted on 3rd party memory cards during the PSX days that I never even took a chance with them once the PS2 rolled around. Now I have ziploc baggies full of them. So many great games. So many save files.
I have a Madcatz PS2 MC and it works great without any issues, but a Chinese non-branded 64MB card works partially due to not having MagicGate (so can't install FMCB on it).
Edit: Also own a bunch of wireless PS2 Madcatz controllers and love them. They're comfortable and work as it should, although the ones that require 3x batteries are stretching it a bit so I prefer 2x AA version.