But since AM, FM and all work in different frequencies, do they have separate types of antennas or similar type of antennas with different lengths?
Generally, you can use the same antenna style and just scale it – that is, until either things get mechanically hard to do, or side effects that you could ignore on one frequency ruin your day.
Also, antennas fulfill different use cases, and that defines what shape they have and what tradeoffs, e.g. in terms of efficiency versus size, are made
Why does low frequency signal have high range but high frequency does not?
That's simply not the case – waves propagate to infinity, and the area power density always drops with the square of distance; see Friis' transmission equation for details.
What's different is that you also get the square of frequency as loss over distance, but fun fact: you also get the square of frequency in effective antenna area per mechanical antenna dimension, so, if you keep your antennas the same size, the effects cancel out (as long as you can use that larger effective area, which usually requires knowing the direction the wave travels).
What people like to forget: a MW broadcasting tower has in the order of 100 kW of power. Your bluetooth device has at most 10 mW. That's a difference of 10⁷, and that alone yields means you get \$\sqrt{10^7}= 10^{3.5} \approx 3160\$ times less reach at the same receive power if everything else was the same.