How could I design a circuit for a solar panel so that it only produces output from a red laser?
Firstly, "laser" tag doesn't (usually) use lasers - and for good reason:

Lasers are dangerous and can blind people. Use with caution.

Laser tag normally operates using a focused infra-red beam. Basically a powerful and focused infra-red remote control (like you have for the TV). This is usually backed up with normal focused white light (from a white LED) for targeting. Yes, a weak laser beam could be used instead of the white light, but it's not the laser that usually carries the information.

You transmit by using a pulsed square wave (the length of pulses and gaps, aka marks and spaces of the signal indicate 0 and 1).

The carrier frequency for the IR transmission varies, but on the laser-tag systems I repair on a regular basis (and have built extra parts for) runs at 50KHz. TV remotes run, usually, at 36KHz (the higher frequency was chosen so that you couldn't cheat by bringing a remote control with you to a game).

The receiver, which is made from a couple of infra-red photodiodes, uses a high-pass (or band-pass) filter tuned to the carrier frequency, along with an amplifier, to isolate the 50KHz (or whatever your chosen frequency is) light signal to recover the on-off pulses. Basic systems don't bother about encoding or decoding information - the mere presence of these pulses over a minimum time frame is interpreted as a "hit". More complex systems include the ID of the gun that has made the hit for score tallying and other statistics.

(Note: this is from first-hand experience of one system alone. There may be other systems out there that do things differently.)