Not really...and I'm going to stick to personally owned vehicles not the commercial vehicle regs which contain all kinds of variants. There is the Transportation Interstate Compact that all 50 states are part of that require each state to honor the registration and many/most equipment regulations for all other states. Now...there are exceptions, but they are so far and few between that the average person will never encounter them.
This Interstate Compact is designed to prevent exactly what the OP described in his initial post. This is why some states have only one license plate/some have two, some have inspection stickers/some don't, and so on and so forth. Texas does not require a driver/owner of a privately owned, non-commercial vehicle to carry a paper document of any kind that describes the vehicle registration information. The license plate and windshield sticker confirm vehicle registration and can be checked by computer if the law enforcement entity suspects fraud or stolen...and of course in the case of a motorcycle, the license plate and plate sticker confirm the registration.
In Texas you're required to carry your DL and insurance confirmation document (proof of financial responsibility). There are some minor exceptions, like for a car that was just purchased and ownership has not transferred...paper dealer tag, just purchased from another owner who is not a dealer, etc. You should have a bill of sale and perhaps even the signed title in hand to confirm the purchase. And there are other minor issues but they are not common.
The Interstate Compact was a very logical agreement between all 50 states...now many decades old...otherwise you'd have chaos. Now, none of this prevents deputy dawg from Blowfly Hollow from writing you up for having green hair and riding a purple Grom, obviously, but his ticket is worthless. I do understand not wanting to go back to that jurisdiction to fight it, but let me suggest this. Call the city, county, or whoever prosecutor's office who would handle this case if you contest it. Tell them you have relatives, business, or whatever in the area of this jurisdiction so you're there frequently and don't mind fighting this silly charge. I'll bet at least a nickel that they will drop that ticket.