The +1 chain shirt is more durable than a breastplate, but this is a very relatively very small difference between them. Consider, for example, that you could have a +1 breastplate nearly as easily as a +1 chain shirt and this becomes quite clear—now the breastplate is again strictly superior.
However, all is not lost—any reasonable DM should be happy to let you describe your character’s visual appearance more-or-less any way you like. If it’s a special, custom item that makes that happen—a superior +1 chain shirt that despite appearances functions as a +1 breastplate, then that’s fine. Or just handwave the issue away. The point remains that beyond the very earliest levels, characters are expected to be wearing whatever armor gives them the best AC, or the best AC without disadvantage on Stealth if that’s important to them. The game is designed for fairly tight AC ranges, and not doing that causes problems. The difference in cost and availability between a chain shirt and a breastplate ceases to matter much at all once you’ve leveled up once or twice. You should neither be “punished” for meeting that numerical expectation (by wearing an armor that does’t jibe with your preferences for the character), or punished for having your character have a certain visual appearance (by having less AC than you should have by all rights). Neither of those results is good for the game—both of them are bad for the game.
Thus, a good DM fixes the problem. That’s why you have a living, breathing, thinking DM in the first place, and not a computer.