Why Sinhalese symbolized by a lion while there is non in the island
King Vijaya is the first king who ruled the country , who came to this island on 486 BC. There was a symbol of lion on the flag he bought from india. From then on this lion symbol played a very important role in Sri Lankan history symbolizing Sinhalese people. This was the start and from there on each and every flag belongs to most of the kings who ruled the country used lion symbol.
But there are many other stories too...
The Lion was not a royal symbol for these ancient monarchs and they used the lion image on foot-stones at entrances to buildings and on urinal-stones. The lion appears to have been an important symbol only for the Indian born Kalinga kings of Sri Lanka, particularly king Nissankamalla (1187-1196 ) source
And some others say there were lions in this island. And many historians found lot of evidence to support that..
The Sri Lankan lion (Panthera leo sinhaleyus), also known as the Ceylonese lion, is an extinct prehistoric subspecies of lion, excavated in Sri Lanka. It is believed to have become extinct prior to the arrival of culturally modern humans, c. 37,000 years BC. This lion is only known from two teeth found in deposits at Kuruwita. source
So these are the most nearest reason why Sinhalese people used lions as a symbol.
Thank you.