If its a grafted tree rose you have, then the growth coming up from below is the Rootstock that the desired variety was grafted onto. Its unlikely to be a variety you'll want, and almost certainly not a repeat bloomer. Make your decision based on that info.

You don't want a sucker. Dump the dirt, clean the pot, and put the rose tree in the trash.

Your friends will hate me for thos but something is growing well in that pot, it's yours and didn't cost anything so keep it just a is and see what it does. I used to cut off suckers and offshoots to reroot but this year just let them be and the quality from that new growth is amazing. When we moved into this house there was a gorgeous Chicago Peace that was soon attacked by something and died all the way down to the ground. It had such big roots that proved impossible to get out to toss so mI just left it for dead and planted around it. In a bit started to leaf and than something gor it again. It has branched out this season from the base with huge, fragrant blooms that far surpass the original. So sometimes it just pays to ignore it and keep on doing whatever or not you are doing. Now I must admit I have three suckers in large pots first year one is in the ground and I bet my neighbor that I could get them to bloom and they did at first but not all that gorgeous like the original and now are just leafing but very healthy so eventually if I need the potted space and they don't pan out not into any great expense. In other words wait and see what the magic of nature brings..

Chicago Peace is just a sport of Peace rose. Peace rose has hardly any fragrance, is not very hardy, and is susceptible to blackspot. Chicago Peace is the same as Peace, just pink. Hardly any fragrance, very susceptible to blackspot. There is no wonder your Chicago Peace died. Just as well, it's a garbage rose to begin with. Just because the sucker is better than Chicago Peace, is not saying very much. I wouldn't grow the Chicago Peace at all. Now, if you said Pink Peace, that would be totally different, Pink Peace is extremely fragrant, hardy, and extremely easy to grow, it won the Gold Medal in 1959, and I would easily grow Pink Peace in my garden if the garden centres would stock it. Since David Austins are being stocked more regularly now, I have a penchant for them, also they are easier to propagate than Hybrid Teas, which makes them more attractive to me. The suckers are just rootstock. You'll have to get rid of it eventually, the sooner the better, so you can grow something worthwhile. May I recommend Tiffany, or Pink Peace, two of my favourites, that have died out from my garden, since I live fairly north. DA's do great here though, they get bigger each year. Sativa.