Not really, but kind of. The bundle of sticks parable was a separate thing for most of the history of the symbol until it was partly merged into the meaning in the 1600s or so. I'd argue even then the difference is cooperation to resist vs the subservience of a peasant empowering the person who oppresses him under the guise of harming their enemies.
"together strong" can be said by any group. Especially fascists, who very much value a sense of community and strength internally.
But really, it can apply to all kinds of governments ("Together for the king!") and economies (like corporation, which is pretty much 'together strong' in capitalist speak).
So I think if one wants to make a point why their system is favorable over other systems, they should not emphasize the one point they all have in common, but highlight where and how it makes a difference.
I'll just assume OP did that IRL. Memes are undercomplex.
The difference is the apes bundle of sticks reinforce each other, to make a larger, more durable unit, while the fascist's bundle of sticks is only there to protect the ax handle. Those sticks are disposable and replaceable for the preservation of the ax, that's their purpose.
There's an episode of the last of us about this, a dude denies his community being socialist and a woman corrects him saying that yeah, by definition they are a socialist community.
I just use an analogy and tell them that at the end of the day, what I believe boils down to this:
If $100 of my taxes is going towards healthcare, then I want $100 used ON my healthcare. I don't want that $100 given to a company that is then profit-motivated to spend $20 and pocket the rest.
I find that those who despise socialism in America...
Not only do not know what it is but have no idea just how socialistic America is. They don't realize that we have extensive socialist support.... For the Wealthy