Greetings! Welcome to the forum... I am just above you in Putnam County... Mahopac / Carmel area..... Most figs in our area will die back to the ground and re-sprout unless they are protected or unless there is a micro-climate protecting them. Thus bush rather than single trunk growing habits usually prevail as a natural state. In any case, without special circumstances, breba crops are not usual.... The fact that you appear to have some on at least one branch is good news and suggests both cold hardiness of the fig and a strong, vigorous rootstock with that concrete block stone wall behind the fig in the picture providing a nice micro climate. I would suggest letting the fig do its own thing for a few weeks until you know what you have and what you plan on doing with it. If you are going to winter protect with covering, etc, you may want to think about how you want the plant shape to be going into the winter and remove any dead wood or shoots that might not align with your thinking / planning for the next cycle.... For example, you could want a single or a limited number of central trunks because they are easier to cover for winter protection... Considering that now and possibly doing some advance pruning will allow you to manage the growth and funnel vigor into the shape you choose.
If you know nothing about the fig... i.e. type, color, etc then the year will be one of discovery... You can lookup and follow some of the fertilization guidelines others have posted on this and the other forum to maximize any potential fruit yields and ensure the plant is in good shape for next next cold cycle. If you do decide to cut away and live wood to better setup the plant for cold protection for next year, then I suggest you do not get rid of the pruned cuttings but either root them for yourself, or if you want, you can offer them to any interested parties on the forum in whatever manner you wish. Since it is an UNKNOWN cultivar but seems very cold hardy, folks will be interested in that characteristic and you will find no shortage of interested folks.... Good luck with this fig, I will be interested in seeing what it looks like and tastes like as the growing year progresses.
Tony