It does seem like splitting hairs to have both. I don't think we are going to have enough specific questions making the distinction between the person and the company to make a difference.
Similarly I would not think we need banker and bank. The one exception might be insurance company vs. agent as they are more different/distinct roles than broker/brokerage in my mind.
The distinction between Broker and Brokerage is important! I do not recommend merging them!
A broker is a person (proper noun), who must satisfy various legal requirements to achieve that status, including testing administered by his employer or the state. This is true in most or all countries. Some brokers are independent, some are affiliated with brokerage firms. The matter of an "unlicensed broker" is not a minor one, and sometimes a consideration for novice investors. (An unlicensed broker would be one whose license has been revoked, or lapsed.)
A brokerage firm is an organization. It too must satisfy a set of legal requirements to be designated as such. Brokerage firms charge fees. Brokers charge fees, sometimes called commissions. Thus I would consider brokerage as an adjective, not a noun. Please see the comment by @JoeTaxpayer on 23 May at 15:53. He is correct! It would be extremely misleading to colloquially refer to the fee associated with a brokerage as brokerage, which was stated as a possibility by @Dheer.
@bstpierre made this important observation:
all of the questions tagged as broker really mean brokerage by your definition
To me, that speaks to the fact that better oversight is needed. If this situation is already occurring, then money.stackexchange.com is going to be significantly impacted in its usability.
I am not overstating the importance of this matter. What is my basis for such confidence, as a StackFamily newbie? First, I am an avid participant in English.StackExchange. (Yes, I'm one of those types. Thank you @JeffAtwood et al for creating the English Stack! It is GREAT fun). Secondly, I possess an M.B.A. with a concentration in Finance from the Wharton School at the University of Pennsylvania, with ten years working experience in that industry. I would not mention this, as it seems like grandstanding. But this is a truly meta question for money.stackexchange.com and this is the best substantiation I could think of, given the perhaps overly emphatic expression of my opinion.
All the comments above pertained to this answer : Is there a difference between tags broker and brokerage Answer 278
The distinction between bank and banker was also mentioned. I agree with @sdg that it is minor. However, I would also agree with @sdg about the distinction between insurance company versus agent. Those two terms do need to be kept separate and distinct, just as broker and brokerage. See details of answer by @sdg : Is there a difference between tags broker and brokerage Answer 277
Please forgive my stridency on this matter.