Not even close. French presses are way larger, holding a can instead of a mug, generally glass, and are pure immersion brewers while aeropresses are immersion/infusion hybrids, giving you way more options. The grind sizes you use are also vastly different: French press grind is coarse to survive the long immersion, while people generally grind for aeropress in between filter coffee and espresso fineness -- roughly what supermarkets sell as espresso fine (which it isn't, espresso fine grind is basically the consistency of talcum powder and spoils within minutes).

And while it wouldn't be right to claim that you can use them to make actual espresso you can use them to make concentrates that come darn close, definitely appropriate for a cappuccino, or tiramisu. You really don't want to make concentrates with immersion.

Oh and by default aeropresses use paper filters, while French presses use sieves. Preferences differ but as you can get sieves for the aeropress again you have more options.

In short, it's the brewer for someone who cares about coffee, probably has a (hand) grinder (and a mere chestnut at that), avoids buying any supermarket coffee and knows a source of proper but non-fancy beans, but doesn't really want to go full nerd about it. Also, isn't a hipster paying through their nose to get a Hario filter holder and papers in a Melitta region (or the opposite), or gets a ceramic filter holder which only means you have to heat it up... no upsides. Speaking of nerds.

In even shorter, it's at a very very solid performance vs. fuss sweetspot. At least if you're making a mug of coffee, if you need to supply a table full of guests... honestly if I had to do it right now I'd throw grinds and water into a pot, wait a bit, then filter the whole thing through an ordinary kitchen sieve followed by an ordinary paper filter holder, and hope for the best.