You can use your Tournaments bikes in Multiplayer and Single Player by going to the Shop and selecting the bike you want to race with!

Just tap on the Multiplayer menu, then click on the shopping cart icon on the right to open the Shop. Scroll left or right to see all the available bikes.

You can only equip bikes you have already unlocked, and you can identify the unlocked bikes by checking the upper right corner of each card. If the button "Select" is displayed in the card, it means that you have already unlocked that bike and it's available to equip it!