Can't believe no one has said this yet:
You learn to switch between bikes by switching between more bikes. Right now you're going between an upright standard and a sumo, and that's the extremes of what you're comfortable with. Go ride a cruiser, go ride an RS125, go ride a big trailie, go ride a WR250X, go ride a Goldwing, ride something vintage, something brand new, a big twin, small 4, big 4, middleweight triple, giant twin, on, and on, and on.
When the range of the bikes you've put some decent time on is everything from a tiny RS125 to a R1200GS, the difference between a DRZ and an SV doesn't seem all that significant. It's all about perspective, maaaaaaan.
Comedy answer: Hop on the new bike, do a wheelie and a stoppie and then back it in and you'll know exactly how the bike handles.
I don't know what to tell you (helpful, right?) but I've had the same question in my head for years. My husband and I have exactly the same street bike, but his is terrible for me to ride--it's set up really differently. I really struggled to learn my 748 (set up for me) and FZR400 (set up for me) and never got it right. Jumped on the R6 and that felt really natural (it was set up for my husband). His Superhawk--so-so. Borrowed some guy's little 400 somethingorother at the track and put down one of my better times the first lap out. It makes no sense. The thing that the "good rides" have in common is that I forget about the bike being under me. I don't know what makes that happen though.
I'd start by looking at the similarities between the bikes you felt natural on - seat position, weight distribution, suspension spring rates, engine power delivery and braking characteristics. I've learned that on the track I like a seating position that isn't available in any modern supersport, it's pretty much a normal standard seating position canted forward about 20 degrees. So handlebars and custom rearsets are the order of the day for me. My preferred suspension setup is closer to Ducati than R6 - a little effort to turn is fine, I want the bike to naturally feel stable. Other folks prefer a bike that turns "telepathically", and is on it's side when you start thinking about the turn. It's just about identifying the consistent bits that make up "natural" for you.