I need to replace this light-bulb. It is very slim and I have never seen it anywhere else. What is the light bulb's type?
There is no doubt that is a halogen tube style bulb. These types of bulbs have a risk to them; they get extremely hot, and can start fires.
Unless the socket can be changed to another bulb type, I advise discarding this entire fixture and replacing with LED. Aside from the higher fire risk, they also are quite inefficient. This 100W lamp replaces about 140W of incandescent, but a 20W CFL will replace it, as will a 15W LED. Your local power company may well have incentives/rebates/bounties for the elimination of such lights, because paying you to get rid of such fixtures is actually cheaper for them than building out the power plant capacity to power it.
My rule of thumb on energy use is $1/watt/year for a load on 24x7 (like that cable box), assuming 12 cents a KWH. A load on a fraction of the time is that fraction of $1. I.E. A bathroom light that might be on 10% of the time, saving 85 watts means saving $85 x 10% = $8.50/year.