That's an edible fig. May not be old enough or watered enough to bloom.

Agree, F. carica. When it does make fruit, you'll need a huge ladder to harvest. I'd start trimming it, and use the pieces you remove to start new ones, not so close to a wall, if you want.

If you train and shape one more like this, you can harvest more figs than you could imagine.

Most likely it is a seedling tree which was planted by a bird (look how close to the wall it is) and still immature. Fig trees are like Citrus and need to reach a certain size before fruiting, multiple trunks apparently delay it further. The lower parts of the trunk will most likely never fruit... But that is not the only problem seedlings have.

At least half are male (capri) trees and are dry and inedible, in some cases all seedlings will be male because the gene for edible female figs is recessive and males can pollinate each other (they are actually hermaphrodite, but considered "functionally male"). http://waynesword.palomar.edu/pljun99b.htm

And then there is also the issue of persistence aka parthenocarpy, where fruit develops without pollination. This gene is fatal to the seed when passed from the female, so it must be passed by the male parent and only half of the seedlings will inherit it as a result.

In socal there are wild male trees which host the pollinator wasp so this is not a deal breaker for fruit, but pollination may be poor if there are no males in the immediate area or they are pruned regularly or something like that.

If the tree is mature but requires pollination figs will form but drop off very small unless they are visited by the fig wasp. They could have been unnoticed.

The good news is that tree can be used as a rootstock for any edible variety to be grafted onto and would be fruiting right away if it works out.

Looks like a parasol tree.

It’s a fig, as established five years ago.