EDIT (5/09/2020) 

Milestone #2: Zelda beats Mario on Switch alone.

My original prediction was that Super Mario Odyssey would be outsold by Breath of the Wild combined Wii U + Switch sales.  Well now it doesn't even need Wii U sales to help it.  Zelda just plain outsold Mario on the Switch alone.  Currently, official Nintendo numbers has Zelda and Mario even at 17.41m, but they list Zelda above Mario on their "Top Selling Switch Titles" page.  I suspect that Zelda is above Mario, but by less than 10,000 units, so they look the same after rounding.

https://www.nintendo.co.jp/ir/en/finance/software/index.html

(end EDIT.)

Prediction:  Lifetime sales of The Legend of Zelda: Breath of the Wild (combined) will exceed lifetime sales of Super Mario Odyssey.

This must seem like crazy prediction.  After all Breath of the Wild was released about a year ago.  Super Mario Odyssey was released just a few months ago, and in that short amount of time, it has shot past Breath of the Wild (Switch) by a margin of about 3m.  Surely there is no way that Zelda can catch up.  Right?  Wrong!  These sales rates are going to reverse, and they are going to reverse soon.  Over the long haul Zelda will continue to sell very well.  Mario will sell ok, but nothing like the first year.

Now some of you are thinking, "But Mario ALWAYS outsells Zelda".  Some may even be thinking something a little more sophisticated like, "First year sales are indicative of lifetime sales."  I am aware of these arguments, but I don't think they apply in this case.  Instead here are my arguments.

1)  A Zelda game like Breath of the Wild only comes around once every few console generations.  The only Zelda games that have had an impact anything like this are Ocarina of Time and the original NES, Legend of Zelda.  Ocarina of Time sales were hindered, because it was a very popular game on a somewhat unpopular console.  NES Zelda was a very popular game on a very popular console, but people who are too young to remember the NES do not understand the NES phenomenon.  NES only had a significant presence in US and Japan.  Switch is very popular like NES, but it has a worldwide presence and also the US population has grown since the NES days.  Taking these factors into account, Breath of the Wild will sell at least 2.5 times what NES Zelda did (possibly more).  The conclusion of this is:

Breath of the Wild will have a minimum of 16.5m in sales.

 

2)  A 3D Mario game like Odyssey would normally not be selling this well.  However it has a very huge thing helping it this time.  It's this:

Mario Odyssey has been a pack-in for the Switch in several places in the world, including Japan.  In Japan, 3D Mario has never been popular, but people are so on fire to play Splatoon 2, that they will buy a game with a pack-in that they don't particularly want.  Of course soon this sort of thing will stop, because of this bundle:

However, my point is that Odyssey will soon start selling like every other 3D Mario game.  It received at most a 3m sales boost by being a pack-in.  So to sum up what I am saying:

Mario Odyssey will have maximum sales of 16m.

So, putting my two bolded statements together, Zelda's minimum, 16.5m is greater than Mario's maximum, 16m.  It's going to be a close race, but in the end Zelda will come out on top.  Of course it will be several years before Zelda actually passes Mario.  Instead what you should expect to see is this:

Nintendo's end of fiscal year, Mar. 31 2018, financial briefing will say Mario sold more (of course).  Then the briefing for Mar. 31, 2019 will say Zelda sold more.  What about Mar 31, 2020?  Zelda on top.  Every year after that?  Zelda on top.  Eventually Zelda will pass Mario and be the better selling game on the Switch.

So, I am saying that this will be like a tortoise and the hare situation.  Mario has shot out in front of Zelda.  But slowly and steadily Zelda will gain ground on Mario, until it finally passes Mario.  By the end of Switch's lifetime The Legend of Zelda: Breath of the Wild will outsell Super Mario Odyssey.


EDIT (5/4/2018)

Here are the sales figures as of March 31, 2018
Super Mario Odyssey: 10.41m
Breath of the Wild: (Switch) 8.48m + (Wii U) 1.5m = 9.98m

Breath of the Wild is only behind Odyssey by half a million.  We'll see how long it takes for it to catch up.


EDIT (5/3/2019)

As always I'm adding in 1.5m to Breath of the Wild sales since that was the last number I've seen for Wii U sales.

If things continue at this pace, then Zelda should pass Mario by this time next year.

EDIT (8/1/2019)

The prediction has been fulfilled!  Here is a short description and my original post is below that.

I originally predicted Breath of the Wild (Switch + Wii U combined) would outsell Super Mario Odyssey.  Last Wii U sales figures reported were 1.5m for Zelda: BotW.  Adding this to the Switch totals yields the table below.

BotW vs. Odyssey sales (in millions)

The Legend of Zelda: Breath of the Wild (combined) has outsold Super Mario Odyssey as of June 30, 2019!

Those bundles were quiet limited and were not exactly a pack in, since the added cost of the game was added to the price.

Wasn't that a limited bundle.

That was very limited bundle and 3D Mario was popular in Japan before.

Zelda deserves it. It is a better game than Odyssey.

Look, I'd love to, but no. Odyssey will breeze to 20m, while your 16.5m for BotW is a tall order, even combined with the WiiU version.

In my opinion: BotW can't beat Odyssey. Mario will have better legs since it is more popular.

I thought this was a Tbone51 thread

I don't think so... but both will have fantastic sales for sure! They'll be the best selling games in their lines.

Mario Odyssey I can see doing 20+ million while I can't see Zelda going much past the likes of 15 million.

A Zelda Botw bundle would sure do crazy numbers though if it gets one at some point.

Zelda’s legs are actually better than Odyssey’s in Japan for some time. But I think Odyssey will easily cross 20m, and that’s a tall order for Zelda.