Are bone breaks in MMA more common nowadays than in the past?

Perhaps! But it really depends what specific time period you're referring to. Just saying "Past" doesn't specify what time period you (or Joe Rogan in this case) are referring to. I'm not gonna provide any numbers to confirm or dispute that hypothesis in this answer.

There have been a number of bone breaks in the past few UFC events, notably the bone breaks of Jacare Souza, Chris Weidman, and Connor Mcgregor. But to truly be able to compare increase of bone breaks (or any statistical event) you need a large enough control group and experiment group. I don't think there's a big enough sample size of the experiment group in this case, just looking at the past couple of pay-per-view events. Whilst this hypothesis may be true, there has been a number of bone breaks to some of the more known names in UFC recently, promoting jumping to conclusions without supporting data. Also need to narrow down what is considered a bone break. Is a broken nose counted too? Or are you just referring to limbs?

Interestingly, Chael Sonnen did a video just on the same topic today. And he pointed out that greatest Jiu Jitsu skill differential existed when Royce Gracie fought in early days of UFC. Although that put him in a huge advantage to break opponent bones, he didn't, as he let his opponents tap under submissions.

There are a few things in my mind to note, comparing bone breaks of now, to the past:

• Need to define "now" and "past" with similar amount of fights between those 2 periods
• Many bone breaks go without becoming a major talking point. Far more common bone breaks, in days past, and present, are those not often talked about. Many fights result in broken noses, jaws, hand, toes, ribs, orbital bones, etc. I doubt there's been a huge spike of such bone breaks overtime, though I wouldn't be surprised if there's been a steady rise in those. I don't think that's what you're asking either, and you're rather thinking of major limb bones.
• The quality of athletes has raised steadily, so the technique and aggression should be at higher levels. This means the defensive skills have also been raised too. So I'm not sure how this relates to bone breaks.
• There has been a number of notable limb bone breaks recently, but how many more in the past 2 years? Yes we've seen a number of those recently, but the sample size is very small.
• There are more fighters that either don't tap under submissions which results in broken bones or torn ligaments (ex: Tony Ferguson), or fights go on far longer than should've due to referees not stopping the fight soon enough or the corners not throwing in the towel (ex: Anthony Smith getting mauled by Glover Teixeira to the point his teeth were falling out)
• The style of fighting goes over changes every few years and also over time. We're in a era where kicks are very common, and they inherently increase odds of injury, including broken bones, specially when checked. Or creating so much pain in a leg that the opponent becomes relatively stationary and more prone to KOs.
• There are far more fights nowadays than decades ago. In all of the 90s UFC had 27 events. But the UFC does that now in just over 2 years, but if you include the weekly fight nights, 27 events happen just over 6 months. So with the increase in number of fights, occurrences of events, such as bone breaks have also increased. But has the rate also increased?
• Recency bias is a thing and we tend to remember recent events in more details then past, and over exaggerate them.

In summary, without proper sample sizes and exact criteria for bone breaks, we can't answer if they've become more common. MMA fights are happening at a higher rate than before, so the occurrence of such event is higher, but we need to look at the rate of such events. If there's a database that keeps track of broken bones in MMA over time, we could study the data and see if there truly has been a raise in amount of broken bones. Otherwise, we can debate and speculate.