The simplest answer I can give is: Yes, Mario Kart Tour is free, but...
To anyone who enjoys this game, that's awesome, but just to let you know my answer below may have some information you might not agree with. That being said, let's go into the pricing/structure of mario kart tour and what to expect.
To anyone unfamiliar with the title, Mario Kart Tour is Nintendo's answer to pushing some of their games to the mobile market. Kart Tour however is not your typical mario kart racing game, it's more of a cash grab "gotcha" scenario where you will be earning coins, tickets, stars, rubies, etc while attempting to have fun with a bare bones mario kart spinoff.
It is a mobile game meant for a mobile market which means that it will have stereotypical mobile game elements. You pretty much use a single finger to maneuver your car throughout the race. You can drift just like with the other kart games and use a variety of different weapons and power ups. Nintendo put in a "pipe mechanic" where you the player have to earn more characters, vehicles, and hang glider type things. Remember those rubies we talked about earlier? Yeah, well you use this game's consumable to fire the pipe and out come unlockables. It costs you 5 rubies per pipe firing.
Now here's where things get extra yucky: three rubies costs $1.99 on the game's store with the option to buy 135 stinking rubies for $69.99 (which is a whopping $10 more than Mario Kart 8 Deluxe on the Nintendo Switch).
You earn stars as you continue to play through the game which eventually will unlock more courses. Stars are given to you depending on how you place in each race, maneuvering and drifting well, using power ups, and getting some sweet hang time off the jumps. Depending on how you place in the races, you can earn more points.
Now for the Mario Kart Tour Subscription:
To be able to play the better/faster version of the MKT game, you will need to shell out $4.99/month. Mario Kart Tour's highest cc option is 200cc and is only available with the subscription service. When a player is subscribed, the way that they receive consumables also apparently changes and you earn more coins, rubies, and game rewards quicker.
I personally don't like this game and it feels like a pretty big miss to me in terms of where the game could have gone. The full game of Mario Kart Tour is so locked behind a paywall that it feels quite lackluster to me. I do know that some people enjoyed the game and I wish I too was one of those, but I guess it's just not for me. Give it a shot if you want.
TLDR: Mario Kart Tour starts off as free to play F2P, but if you want to access the whole game, you will have to pay for the subscription service which is $4.99 a month.