Leaves are much too large for Siberian elm. It looks like it could be American or oriental hornbeam, too.

".... could an American elm survive in a pot on a condo balcony/patio?" Yes. Make sure the pot is against the exterior wall of the building. Head conducting through the walking surface of the patio and exterior wall would be beneficial. Your tree needs a cold rest, so you shouldn't consider trying to 'protect' it by bringing it indoors.

"The plant identification app told me it’s an American elm, which are big trees. Is there anything I can do with this one?" Little trees become big trees only if you allow them to become big. I have several species of elm growing as bonsai, as well as oriental and American hornbeams with leaves of similar size. All you need to do is prune it back annually after leaf loss and before the spring push. Sooner is better to prevent sap weeping from pruning wounds in the spring, which tends to attract insects that feed on sap.

"I’ll try to move it to its own pot when its leaves fall but what are its prospects?" Prospects are very good. It's a naturally vigorous tree and will grow with vitality if it's cultural needs are met. If you intend to repot it, you should wait until spring. Keep an eye on when the beds start to swell and repot it at the first hint of bud movement. Note the date of bud movement and repot 2 weeks earlier for subsequent repots. The plant will need regular root pruning - at least every other year is best, but in a pinch it could probably be repotted every third year but the tree will lose significant growth potential in the second and especially the third year after the most recent repot.

"When will it become unmanageable?" If you prune the roots and top judiciously the plant will remain manageable as long as you remain able-bodied.

This is a zelcova ^^^, a member of the Ulmus family and a close relative of elms. Leaves are about the same size as yours, but can be reduced to about 1/3 the normal size (or less) with bonsai techniques. In the image, the plant is about 12 years from a cutting, the trunk about 4.5" in diameter at the base. Today the trunk is about 6" in diameter at 15 years from a cutting. It's not in a bonsai pot yet because I'm waiting for the wound to close where the trunk was originally chopped to produce taper toward the top, and the larger soil volume of the cut-down nursery can speeds the healing process. Right now, I'm wrestling with air layering the top off immediately below the wound to produce a much shorter tree with rapid taper (adds to the illusion of great age). Then, I would use the rooted stump to grow a broom style tree like the zelcova below. Exciting stuff to a bonsai practitioner.

As you can see, there are many options; and, if you ever feel the tree has outgrown the caregiver, call a bonsai club. As long as there is some substance to the trunk, there will be someone very happy to get the tree - no matter what the top looks like.

"Should I just throw it away now and not bother or it could be useful for an urban patio in a container?" That's entirely your call. Elms tend to get ratty toward the end of summer, but they are so vigorous you can defoliate them in the first week of July (where you live) and they will put on a flush of new, pristine foliage within a few weeks which should remain in good shape until it's time to start preparing for its winter rest.

Consider the discovery of your volunteer tree to be a case of serendipity. You could hardly ask for a better species to experiment with.

Al