Using enemy flags to signal surrender was not the typical method of surrendering, the Russian fleet resorted to using Japanese flags because the more traditional methods of surrender were not accepted by the Japanese.

First the Russian fleet did strike their colours, however some of the flags had already been shot off, during the fighting and therefore the Japanese didn't realise this was an attempt to surrender.

Then the Russians tried to send a message of surrender using phonetic flags. However the Japanese didn't always use the same phonetic codes, so they didn't understand the message.

Then the Russians improvised and used tablecloths as white flags to signal surrender. However Togo, who commanded the Japanese force had previously been tricked by his opponents using white flags during the Sino-Japanese war 10 years ago, so he didn't accept this as a sign of surrender.

It was only after all of this that the Japanese flags were raised. The flags were presumably on the ship to be used as courtesy flags. In maritime etiquette it's customary to fly the flag of the nation whose waters you are sailing in as a courtesy flag. The courtesy flag should hang from the senior signalling position.

Courtesy flags however are only used during peace time, so using them in this instance was enough to signal surrender.

In summary ships would carry the flags of nearby countries to be used as courtesy flags during peace. However they would not have used it to signal surrender. The use of courtesy flags to signal surrender is due to circumstances unique to the battle, and wasn't a common occurrence.