Cyclemeter does do that! You can go to Settings>Bike and add however many bikes you want to the list. From the Stopwatch (main) page, tap the little gear next to the Start bar, go down to Bike and select which bike you are using for this particular ride. When you end your ride, selecting the ride from the History page, Calendar or anywhere else that ride data can be displayed, will show the bike used on that ride on the Info screen. Once you place a bike in the list, Cyclemeter will keep track of all the rides you have done on the bike and the number of miles you've ridden the bike. If you enter the data when setting up the bike info, it will also show you the date the bike was put into service or the date you added the bike to the list and it's wheel circumference; which comes in handy when setting up new bike computers.

If you haven't used Cyclemeter since 2018, you may want to reconsider using it again. They have done quite a lot to the app since then. A new feature that was just released is the ability to trim your ride from the app without having to go to a website to do it. The Trim feature allows you to remove part of your ride in the event you forgot to Stop the ride and the GPS kept on tracking while you're driving home. As a beta tester, I can tell you that we are testing a new feature to Cyclemeter that no other cycling app has and that none of the Garmins computers have. I can't tell you what it is but it will be worth while waiting for.