Not having any idea what the variety is will make it difficult for a while, especially as it's produced flower so small. The habit will quickly reveal itself, however, when it produces a bit of new growth from the base. I know the flowers are pretty, but cutting it back now won't harm, and will trigger it into new growth which may just give you a clue because of its vigour and the extensions between nodes. If it were me, I'd just wait, look after it, enjoy the flower, and let it do what it wants to! Even if it is a bush/shrub rose, it can still easily reach the top of a fence.
Some roses, as you probably know, can be bush or climbing (e.g. Iceberg), and there's no obvious difference between a small climbing Iceberg, and a tall bush form. Doubtless there's an expert on roses here somewhere who'll be able to add clues or even identify the one you have. To me, from the flower type, it looks like a bush rose.