Berry growth seems stalled in Pokémon Emerald. Is it possible I have a fake cartridge?

Solution 1:

I'm sorry to say, that cartridge is fake.

As you rightfully pointed out, the black blob you have was never used on official Nintendo cartridges.
Some other indicators are; the Nintendo logo uses the wrong font, there are almost no white silk-screen markings aside from the logo, the holes along the top of the pins don't line up, and as you also noted it has no battery.

I can't tell from your photo, but all official GBA cartridges have a 1 or 2-digit number stamped into the sticker, and I've never seen a fake one with one.

The reason your berries are not growing is due to the lack of a battery for the real-time clock. Without it, the game has no way of knowing how much time has passed since you last played. The official game has a warning at the start saying that berries won't grow if it doesn't get a reading from the clock, but I've heard of some fakes removing the warning from the game code.

There are many good resources for identifying fake games, here is one that is relevant:
https://caggames.weebly.com/how-to-spot-a-fake-pokemon-gba-game.html