Futures contracts have specific dates so as to concentrate all liquidity into a small set of identical deals. This is more efficient for the market and leads numerous effects eg, tighter spreads etc.
OTC fixed income is well, OTC. So it can be whatever you want and participants often want to specify exactly how the cash flows fall, as this enables them to balance cash flows they have due to other business.
An important distinction to note here is that futures exchanges don't directly offer risk. They operate the exchange, but another customer is always on the other side of your trade. In other words, the exchange does not take on the other side of your deal unless it has found someone to do the opposite with. Therefore, custom deals would be basically impossible to match at scale.
In OTC markets, when you want to trade, you ring a broker / bank who creates a deal for you and assumes the other side directly. So in practice they can structure the cash flows to whatever you need (assuming the prices is right of course). This means that banks end up with thousands of kinda offsetting deals with hundreds of clients, and yes it's a mess. Sometimes two banks have tens of thousands of deals with each other, and they compute the sum and agree to one new summation deal and tear up the old ones.
Additionally, there ARE many market 'conventions' in the fixed income and fx OTC markets that do make things easier / faster / more liquid. For example, many market participants agree to use the same set of dates for forward starting FX and IRS, called IMM dates. So if you ring the bank and ask for "Dec IMM forward starting USD 10y vanilla IRS", and get "14 / 20' and you reply "mine", well you've done a deal and it's likely you'll agree on what that exactly was.