TL:DR in spoiler (hopefully simplified)

Edit: Alternate second method using official PS3 figures as of March 31st 2017 and breaking down PS3 FY sales to determine PS2+PS3 combined possible minimum split. Thanks to @RolStoppable for linking the PS3 march 2017 figures. This method shows PS2 HAD to have sold over 158.29m (Link)

The last official number for PS2 ever given was 155.01m as of March 31st 2012. The PS2 was discontinued in January 2013 and we never got another number because starting with their quarterly report for June 30th 2012 Sony began combining PS2+PS3 numbers.

However because we did receive PS3 shipment numbers in the following year we can use the combined numbers to determine a range that the PS2 must have ended up at. Here's what we have. (All sources will be linked at the bottom of the post)

March 31st 2012
PS2 = 155.01m
PS3 = 63.9m

June 30th 2012 - PS2+PS3 shipped 2.8m this quarter
September 30th 2012 - PS2+PS3 shipped 3.5m this quarter
December 31st 2012 - PS2+PS3 shipped 6.8m this quarter
March 31st 2013 - PS2+PS3 shipped 3.4m this quarter

So the total combined sales of PS2+PS3 for the year of PS2 sales we have missing is 16.5m. Now how do we work out how many of those 16.5m are PS2's?

Well luckily we have some help, Sony announced that as of November 2nd 2013 the PS3 had shipped 80m units. This means that between March 31st 2012 and November 2nd 2013 the PS3 shipped 16.1m units, and we have figures for some PS3 quarters in 2013.

June 30th 2013 - PS3 shipped 1.1m this quarter
September 30th 2013 - PS3 shipped 2m this quarter
December 31st 2013 - PS3 shipped 3.3m this quarter

Here's a table with every number we have:

Some of the December 31st 2013 numbers are after the November 2nd date but just taking the June/Sept 2013 quarter numbers away from 16.1m we already bring the missing PS3 shipments down to 13m.

So now we have:

PS2+PS3 shipped 16.5m between April 1st 2012 and March 31st 2013.

PS3 Shipped 13m between April 1st 2012 and March 31st 2013 combined with October 1st 2013 to November 2nd 2013.

So the PS3 shipments in the period where PS2+PS3 shipped 16.5m combined (April 1st 2012 and March 31st 2013) MUST be less than 13m.

The only missing piece of the puzzle is how many of the 3.3m PS3's shipped in the holiday quarter of 2013 were shipped before November 2nd. If we had that or we had PS3 shipments as of December 31st 2013 we could get the exact number of PS2's sold lifetime.

As we can't get that then worst case scenario for PS2 all of PS3's 3.3m holiday quarter shipments in 2013 were after Nov 2nd, meaning PS2 sold 158.6m

Best case scenario for PS2 all of PS3's 3.3m holiday quarter shipments in 2013 were before Nov 2nd, meaning PS2 sold 161.9m

Obviously both extremes are unrealistic, October will generally ship less than Nov/Dec but rather than doing any guesswork and keeping this entire post purely mathematical and void of any guesswork. 

PS2 sold somewhere between 158.51m and 161.81m.

Sources:

PS2+PS3 combined figures (Page 8) - FY 2012 Consolidated Financial Results (sony.com)
PS3 2013 quarterly shipments (Page 13) - FY2013 Consolidated Financial Results (sony.com)
PS3 ships 80m as of November 2nd 2013 - PlayStation®3 Sales Reach 80 Million Units Worldwide (prnewswire.com)

(PS3 final 2013 quarter is determined by taking away the 4.5m shipped PS4's figure from the combined PS3+PS4 sales in the PDF)

Edit: I was digging for information for PS2 shipments because I was requested to make a ps2 vs ds graph, but of course many people have come to this conclusion before me lol.

There's a whole thread discussing the situation in 2018 here: What was the lifetime sales of the PS2? Not 158m [Update: Between 159.2m and 161.9m] | Page 2 | ResetEra

There are people coming up with the same range as me 158.51-161.81m, ZhugeEX says FY12 was probably up YoY and that would increase the minimum to 159.2m

While there is easily some wiggle room for a margin of error of a few million, it would be even more incredible to think if the PS2 reached about 160-161.9 million. As you said, the last official number was around 155 million. And it seems like this site and other educated estimates put the total around 157 million.

Yea there’s been some sources quoting a 158 million number for ps2. Which is believable.

If it sold that much between March 2012 and January 2013 then why would it be discontinued? Consoles usually get discontinued when they sell less than 1 million a year not up to 7 million in 10 months

Sony switched from reporting manufactured units to units sold to retailers as of 2007. So if you look at 2007 or 2008's FY report it will contradict previous reports with slightly different numbers.

This is what we have from Sony:

Manufactured

FY99 - 1.41m (Source)
FY00 - 9.2m (10.61m total) (Source)
FY01 - 18.07m (28.68m total) (Source)
FY02 - 22.52m (51.2m total) (Source)
FY03 - 20.1m (71.3m total) (Source)
FY04 - 16.17m (87.47m total) (Source)
FY05 - 16.22m (103.69m total) (Source)
FY06 - 14.2m (117.89m total) (Source)

Sell-In

FY06 - 14.71m (Source)

FY07 - 13.7m (131.59m total) (Source)
FY08 - 7.91m (139.49m total) (Source)
FY09 - 7.3m (146.8m total) (Source)
FY10 - 6.4m (153.2m total) (Source)
FY11 - 4.1m (157.3m total) (Source)

Edit: I have removed a lot of this post to avoid misleading conclusions but kept the figures for those that are interested. The 155.01m figure seems to be correct as I found this from ZhugeEx: "PS2 Production Shipments were 117.9 million as of March 2007. At this point Sony switched from reporting production shipments to hardware sell in. Hardware sell in at this point was 115.6 million." from this thread

This will bring total Sell-In to 155.01m as of March 31st 2012. It's difficult finding information on this from a decade ago as 155m is quoted in many places but never given a source or rationale on how this fits into the fiscal data supplied by Sony, but here it is. I will try and find where this 115.6m figure comes from but I doubt I will have luck lol.

Are we making sure to establish accurate PS2 sales in case the Switch gets there? Haha...mostly joking, but more hoping.

Well it would be unfortunate if the Switch ends up in the range of "we have no idea if this is higher or lower than PS2" so hopefully if it does get to the 155m mark it blasts a fair bit past 160m too.

Yes, I have explained this many times in the past. It is still difficult to get a solid number. But the gist is, if PS2 didn't sell 160m, then the PS3 did sell closer to 90m than we thought.

Because by January 2013 they were already (or getting ready to) making the PS4. It wouldn't make sense to keep making the PS2 much longer, and it wouldn't make sense to discontinue the PS3 already in 2013 and keep making the PS2 for another few months to a year. 

PS2 only held on so long anyway because of what a titan it was. If the PS3 did better, PS2 probably would've been discontinued in 2009 or 2010. 

I also feel that PlayStation fanboys want to be prepared in case the Nintendo Switch starts to come close to PS2 numbers. I think the Nintendo fanboys could counter that by saying that Nintendo is always slightly lowballing with their official numbers and it ads up over the years. For whatever reason Nintendo keeps their official numbers a bit too low... meaning that at the end of Nintendo Switch's life you probably can ad a few millions.