Capacitor charging not working according to t = rc
Your understanding of the capacitor is correct, your understanding of the power supply is wrong.

At initial connection the cap will try to draw 7.3mA through that 1K resistor assuming the voltage is a solid 7.3V.

Unfortunately, the solar panel can only supply 1.9mA.

On it's own, at that level the capacitor will take a lot longer to charge. (>28S)

However, what also happens is the voltage at the panel will fall since it is being over-loaded. That further reduces the charge current extending the time even longer.

In other words, basically, the solar panel is presenting a large terminal resistance at the currents you are demanding, which affects the time constant.

How you are illuminating the solar panel also matters. Depending on the type of lamp, there may be a significant AC component to the light. That can cause the solar panel to be "dark" for a fraction of the cycle, at which time, it acts as a diode shorting out your circuit.