Antenna gain and aperture
The receiving and transmitting patterns of a antenna are the same.  That's a fundamental property of antennas from physics.

So yes, your two antennas with the same gain, pointed at the same transmitting antenna will received the same strength of signal.

You mention that the first antenna is very thin, so you wonder how it can handle the same power.  It can't, but that is independent of the fraction of power from the transmitting antenna it will receive.  Generally receiving power is very low, so this doesn't usually matter.  However, the total power the first antenna can handle before it melts or whatever is lower, but it will still receive the same power until then.

This is the same as a 1/4 W 10 k&Omega; resistor used in a 5 V opamp circuit acts the same as a 2 W power resistor of 10 k&Omega;.  You only see the difference when you try to put more power into the resistors.  Up until 1/4 W, you can't electrically tell the difference between the two.  At 2 W, the first resistor will melt, vanish into a cloud of greasy black smoke, or whatever, while the second resistor keeps on going.