The post looks reasonably adequate as it was originally built even by modern standards. If it wasn't you'd have seen disaster long ago, when it first started to decay.
To maintain the style detail I'd rebuild to match, using pressure-treated lumber. By doing so you eliminate the need to cut that notch and you end up with a more robust post. A single post member may have weaknesses that are addressed by using multiple boards. To be clear, each board should be cut accurately to fit its bearing points on each end. It's not just the center board carrying the roof load.
I'd fit everything and screw it all together at 18" intervals with pairs of 3" corrosion-resistant deck screws from each side. After it dries out for a few weeks, use a high-quality wood filler to flatten out the screw heads, then prime and paint.
Back in the dark ages, when playgrounds were built with concrete and asphalt surfaces, I saw 2-post swing sets built with 4-6" diameter steel posts (I was a kid then, I never measured accurately). If you want to use steel and weld it up, you're probably good.
If you want to use wood, you'll probably have too much flex. One of those posts could snap after some use (not day one) and send a kid flying. Flying off a swing at the top of the arc is great fun (even at my age), but having it be an unplanned event, with large chunks of wood following, would be considered by most to be a Very Bad Thing™.
Consider both swings going in sync - you've got all the weight pulling on both posts in the same direction at the same time. Now consider both swings going in opposite directions - each post is going to bend in the direction of the local swing, and the cross-member is going to twist. If your kids are even remotely normal, as they get older (and heavier), they will want to see how far they can get the thing to bend by swinging in unison, and how much they can make it twist by swinging in opposite directions.
Bear in mind, your kids may be small now, but they'll grow. We had a small steel swing set in our backyard that we installed when our kids were small. They continued to use it into middle school when they were in the 50-100 lb range. They continued to use it in high school when they were 100 lbs+. They had friends over and had more people on it than it was designed for. They had people swinging on it and hanging on it and climbing on the top bar all at the same time. As they hit middle school (and even earlier) they thrilled at trying to make the A-frame swing set lift up by swinging in unison. It came with screw-in anchors. We had to buy more anchors and bigger ones and put one on each of the 8 legs to keep them from tipping it up. It's just what kids do.
No amount of parental policing will ensure that the swing set will only be used according to today's design standard, and even if you manage that, there will be kids sneaking in when nobody's watching, mostly because you've gained a reputation for being the "swing set Nazi" and they want to show you!!! It's just what kids do.
After 10 or 15 or 20 years of rotting weathering and hard use by heavier and heavier kids (we took ours down just a couple of years ago after 2 of the 3 kids moved out - they would still swing on it with their college aged friends when they had them over), even pressure treated lumber will start to wear. There's really no telling when it might give way. Wood's a natural product and the average strength of a PT SPF 6x6 might be sufficient, but the particular pieces you pick up from the lumber yard may be below average (or may be well above average, who knows).
Spend the extra money and yard space for an A-frame design. You don't want the neighbor's kids going to the hospital and their parents suing you.
Best Answer
4x6 is pretty rigid, and doubling them will help, you will need a deep footing though as there will be high repeated lateral forces on it. 3ft is probably not deep enough.
For this reason most swtngs have A-frame ends instead of plain verticals.