It happened, but it was considered a risky procedure and was not popular in the medical establishment. Dracula was published in 1897, three years before Karl Landsteiner discovered blood groups. It wasn't until around 1910 that group matching became common practice and the era of modern transfusions began.

The first reported blood transfusion was performed on dogs by Richard Lower at Oxford in 1666. The next year, Jean Denis in Paris was the first to transfuse animal blood into humans. These were in the nature of experiments; it was not medical practice.

The first human to human transfusion was done by James Blundell in London in 1818. Blundell was an obstetrician who often saw women bleed to death after labor in postpartum hemorrhages. He gave a woman a transfusion of blood from her husband, and she survived. This was encouraging, so he developed techniques and instruments to do it more easily. Aside from not knowing about blood groups, another big problem was clotting. Anticoagulants were unknown at the time, and it was not until 1915 that sodium citrate came into popular use as an anticoagulant in transfusions.

But Blundell knew that one way to decrease the danger of clotting was to speed things up and minimize the time blood remained outside the body. He devised a syringe with two heads and a valve lock, so you could do a phlebotomy on both donor and recipient in advance, draw the blood, flip the valve and then immediately transfuse it to the recipient. Of course, he still didn't know about blood groups, so incompatibility was a problem. It remained a risky procedure, but the patients were dying anyway from lack of blood, so that was some justification.

Blundell produced many devices for facilitating blood transfusions and had them manufactured and sold, and became a rich man from these sales. So apparently there was market enough among other physicians experimenting with blood transfusions during this period (between the 1840's and the time Dracula was written). However, it was still very much an institutional thing, with some small number of hospitals where it was popular while the vast majority of physicians wanted nothing to do with it.

The Edinburgh Royal Infirmary was one of the early institutions where blood transfusions became popular, and in the decade before Dracula they did a lot of transfusions which were widely reported in the press. Perhaps that's where Bram Stoker got the idea. He didn't seem to understand the mechanics though, because van Helsing's technique described in the book is very crude.

Physicians of the time were certainly more sophisticated than that. Landois published a treatise in 1875 - Die Transfusion des Blutes - in which he reported that mixing blood cells from one species with serum from another often causes lysis in 2 minutes or less. This work was in fact the basis of Landsteiner's experiments which led to the discovery of blood groups. But ever since Landois' text, people knew that the incompatibility reaction was quite rapid, so it was standard practice in some institutions to start with a small transfusion and continue a few minutes later only if there were no adverse reactions.