This is a great question! I haven't been able to find a definitive answer, but it seems what Bram Stoker probably had in mind was essentially a light bulb connected by wires to a battery. (This could neatly explain why they were to be "fasten[ed] to your breast": the bulb itself would be quite light.) Certainly it was not a flashlight: the first flashlight with a now-familiar form factor, that is, a tube with a bulb at one end, was marketed in about 1897 or 1898 in the United States. It followed closely the introduction of "D" cell batteries in 1896, which themselves were a major improvement, convenience-wise, to the first portable dry cell batteries invented by Dr. Carl Gassner in 1888.

A flash light historian (!) Stuart Schneider explains that D cell batteries were used to fashion novelty lights:

One of the early electrical novelties powered by a battery was a simple stick pin with a miniature bulb. Wires connected the bulb to a battery hidden in a pocket or behind a scarf (tie). When the wearer pressed a switch carried in the pocket, the bulb flashed. "D" batteries powered the scarf pins. The scarf pin was a novelty when introduced**, but users discovered practical uses for it, such as reading in dark restaurants or theaters.**

(From http://www.wordcraft.net/flashlight.html, emphasis mine)

Stoker might have had in mind a light powered either by a D cell or earlier "#6" battery, which was good enough for useful amounts of portable light, but weighed about 1.5 kgs. While I haven't found direct evidence, it seems to me more than likely that many people constructed contraptions like the one described above to provide some portable light. Stoker wrote novels that are considered to be early examples of science fiction, and he was known to be keenly interested in scientific and technical progress, so the odds are very good that he had seen, or perhaps even built himself, a gadget like that.

In addition to Schneider's work, I consulted

Bunch, Bryan & Hellemans, Alexander (2004). The History of Science and Technology. Houghton Mifflin Company.

So Bram Stocker was actually very well versed in the latest technological developements of his time, because he was for more than 20 years the collaborator of Henri Irving, one of the most important theatre director of his time, working at the Lyceum Theatre. Irving was a pioneer in terms of lighting, en special effects, and we know this in part because of Stocker, who wrote an article called Irving and stage lighting in which he describes the many experiments Irving did witch lighting. Irving was, among other things, a pioneer in using many colors : he invented lacquers he could use to paint the lenses of his lamps, so as to get an infinite number of shades.

What is really relevant to your question, however, is that Irving also did some experiments with electricity to get "special effects". Stocker gives an exemple of that with Faust, in which Irving used electrical sparks to emphasize the sword duel :

Two metal plates were screwed on the stage, to either of which the current of one pole was applied. One of the combatants had a metal plate screwed to the sole of the right shoe. From this a wire was carried through the clothing and brought into the palm of the right hand, where, on the rubber glove, was fixed a piece of metal. This being in contact with the metal handle of the sword – and a similar contrivance being arranged for Mephisopheles – a direct communication was established so soon as the demon’s sword struck up the weapons of the combatants, and sparks were emitted.

This way in using electricity was somewhat new, altough Wagner, in Germany, also used a similar technic. So, this text shows us that Bram Stocker was privy to the latest theatrical experimentations with electricity, that were at the time going on all around Europe.

And one of those experimentations (which was not, to my knowledge, used by Irving himself, but quite famous on the stage nonetheless) were "electrical jewels". Those were first invented by the engineer Gustave Trouvé in the 1860s, but they grew in popularity in the 1880s. They consisted of a small incandescent light bulb that is linked to a small battery. The light bulb is fixed to a person who can light it up when convenient. Here you can find an illustration of the effect those "electrical jewels" had when a dancer would wear them (the image in from 1884). It seems to mirror exactly the description given by Stocker in chapter XIX : "The light from the tiny lamps fell in all sorts of odd forms, as the rays crossed each other, or the opacity of our bodies threw great shadows." Here you have a technical drawing of the mechanism (from the same source). Those were a french invention but also used on the english stage, for exemple in 1884 in the ballet Chilperic, at the Empire Theatre. It is quite likely Stocker would know of them. Trouvé then invented a lot of different "electrical jewel", which were not reserved for the stage. He made lighted hair pin, lighted cravat pins, lighted canes, etc. I do not know if those were really succesful as items to wear outside the stage. However, it seems likely that they could have fuelled Stocker's imagination, since he knew so much about theatre lighting.

So, while I can't say definitely that Trouvé's inventions where exactly what Stocker had in mind when writing that chapter, it is more than likely that he at least pictured a somewhat similar device.

Related question -- Does Hans Christian Andersen refer to the same "electric lamps which you can fasten to your breast" in his story The Days of the Week? (Published 1872, emphasis mine)

"Sunday, foreman of the days of the week, appeared in a black silk cloak; pious people thought he was dressed for church in a minister's gown, but the worldly minded saw that he was attired in a domino for merriment and that the flashing carnation he wore in his buttonhole was a little red theater lantern on which it said, 'All sold out; see now that you enjoy yourselves!'"

https://andersen.sdu.dk/vaerk/hersholt/TheDaysOfTheWeek_e.html