No, it would be a ridiculous request (were you not paying attention?). You should have taken written/typed notes. In particular brief details on questions asked, information about the position, skills, buzzwords, hiring manager, team, hiring process, timelines, compensation etc. Writing stuff down is like 5000 years old. You have no excuse whatsoever for being sloppy. Gives a bad impression, like you weren't taking the interview seriously.
Before any phone interview, try to do all of the following - should only takes 15-30 minutes:
a) print off the job spec and make notes on it / or the back of it (compact, and saves trees)
b) check the interviewer's name and background (so you can predict what they might ask you, and read up on it. Algorithm questions? Behavioral questions? Your resume? Knowledge of the sector?). Always try to get the name of the person who will interview you. If the person you speak to doesn't know it (or doesn't care), ask in the email you send confirming your interview date, time and phone number, or else failing that, call them and ask the name of the interviewer and their position.
c) take out a notepad and pen, write the date & time, name of interviewer, position, phone number, yada yada. Keep those notes for your next round interviews. (Sometimes you will even pick up on discrepancies in the hiring process or the company's internal communication, which might or might not matter)
d) open a vi/emacs/GoogleDoc/task-manager/whatever file to take notes. To the people objecting about typing while on the phone, mute the line whenever you're not speaking, and/or use a headset so it only catches your voice not all teh noise on your end.
e) if you have time to kill, read the company's other job listings, product descriptions(/API documentation), whitepapers, press releases/filings, press coverage, (independent credible) reviews of products/services, strengths and weaknesses, who are their customers, sector reviews, webinars, Q&A, Glassdoor etc. Don't waste time reading boilerplate endorses or PR. Identify major competitors and who they lose(/win) staff or customers to(/from). Identify people you know who can give you an impartial opinion of the company's product, people, positioning, customers, future etc.
Write down important stuff even if you're pretty sure you'll remember it: why? you might forget, as in this case. Other urgent things might come up and disrupt your schedule and concentration. Also so you can review the whole process later in the future, or in case someone you know is going to interview there or some similar company and position.