I'm very specific about flag. I know about the symbolism of bhagwa
color.
If you are only looking for the importance of flag in hinduism then yes they are important.
Many Gods are often associated with particular flags.They are described as having certain flags with specific symbols on them.
Following are few such examples.I have here used the gayatri mantras for several Deities:
Goddess Chamunda is depicted as having a flag with the symbol of "Pisacha" on it.
Om Pisacha Dhwajayai Vidmahe soola hastaya Dhimahi Tanno Kali
Prachodayath.
Similarly Goddess Brahmi is described as having a flag with a swan(hamsa) symbol on it.
Om Hamsa Dhwajayai Vidmahe Kurcha Hastayai Dhimahi Tanno Brahmi Prachodayath.
Goddess Indrani is having an elephant(gaja) on her flag:
Om Gaja Dhwajayai Vidmahe Hastayai Dhimahi Tanno Indrani Prachodayath.
Maheswari-Having a flag with bull on it.
Om Vrisha Dhwajayai Vidmahe Marug Hastaya Dhimahi Tanno Roudri Prachodayath.
Similarly Goddess Varahi is having a Mahisha(buffalo) on her flag etc.
Each of the Navagrahas is also having a particular flag with certain special symbols on them :
Sun
Om Aswadwajaya Vidhmahe Pasa Hasthaya Dheemahe Thanno Surya
Prachodayath.
Om, Let me meditate on the god who has a horse flag, Oh, God who holds
the rope, give me higher intellect, And let Sun God illuminate my
mind.
Moon
Om Padmadwajaya Vidhmahe Hema roopaya Dheemahe Thanno Chandra
Prachodayath.
Om, Let me meditate on God who has lotus in his flag, Oh, God of
golden colour, give me higher intellect, And let moon God illuminate
my mind
Mars
Om veeradhwajaaya vidmahae vighna hastaaya dheemahi tanno bhouma
prachodayaat
Om, Let me meditate on him who has hero in his flag, Oh, He who has
power to solve problems, give me higher intellect, And let the son of
earth God illuminate my mind.
Budha
Om gajadhwajaaya vidmahae sukha hastaaya dheemahi tanno budha:
prachodayaat
Om, Let me meditate on him who has elephant in his flag, Oh, He who
has power to grant pleasure, give me higher intellect, And let Budha
illuminate my mind.
Guru(Jupiter)
Om vrishabadhwajaaya vidmahae kruni hastaaya dheemahi tanno guru:
prachodayaat
Om, Let me meditate on him who has bull in his flag, Oh, He who has
power to get things done, give me higher intellect, And let Guru
illuminate my mind.
Shukra
Om aswadhwajaaya vidmahae dhanur hastaaya dheemahi tanno shukra:
prachodayaat
Om, Let me meditate on him who has horse in his flag, Oh, He who
has a bow in his hand, give me higher intellect, And let Shukra
illuminate my mind.
Shani
Om kaakadhwajaaya vidmahae khadga hastaaya dheemahi tanno mandah:
prachodayaat
Om, Let me meditate on him who has crow in his flag, Oh, He who has a
sword in his hand, give me higher intellect, And let Saneeswara
illuminate my mind.
Rahu
om naakadhwajaaya vidmahae padma hastaaya dheemahi tanno raahu:
prachodayaat
Om, Let me meditate on him who has snake in his flag, Oh, He who has a
lotus in his hand, give me higher intellect, And let Rahu illuminate
my mind.
Kethu
om aswadhwajaaya vidmahae soola hastaaya dheemahi tanno ketu:
prachodayaat
Om, Let me meditate on him who has horse in his flag, Oh, He who has a
trident in his hand, give me higher intellect, And let Kethu
illuminate my mind.