No.
Both different platforms, different fudning (flu is not being moved to new platforms because research funding isn't there), and different rates of mutation and strain change.
Plus the influenza virus mutates enough each year that the vaccine for one strain often provides no or only partial protection for another strain. Part of why "guessing right" about which strains will be the predominant ones in any given year is so important and has such an impact on how effective the vaccine is in any given year.
As for a "cold" vaccine... the common cold is caused by any of about 200 different viruses, only 4 of which come from the coronavirus family. There are other "families" of viruses that also contribute to what is understood as the "common cold" [eg rhinovirus]. So far researchers haven't found a common underlying element that would let them design a vaccine that could be effective against them all.
And viruses can sometimes be tricky to create vaccines for in the first place. HIV is an example. While there are many reasons why there is not an effective vaccine for it yet, one of them is that the human body does not develop good enough immunity to the virus -- including to vaccine versions of it.
For other coronaviruses that affect humans, there was work being done toward vaccines for SARS and MERS. Most of the SARS work stopped when the SARS pandemic ended. However, a lot of the work that was done towards those vaccines was able to be pivoted and used toward the SARS-COV2 [covid] problem, thereby greatly accelerating the timeframe involved [ie years of work that would normally be needed if you started "from scratch" didn't need to be because it was already done and you weren't actually starting from scratch]. Likewise work done towards the Ebola vaccines got pivoted and applied. If that work didn't exist and those vaccines didn't exist then they would be a lot further behind where they are now.
One of the biggest factors though is what Auroraborealis noted: funding. BILLIONS in funding has been thrown at the vaccine efforts for covid. Development work. Clinical trials. Manufacturing sites are being paid for, built and operated in record time and are or will soon be producing vaccine, without knowing if the vaccines will actually be successful. Pre-purchasing of doses by countries who can afford it, and global initiatives to pool donated humanitarian funds to purchase doses for countries who can't.
The closest you see that is anything remotely similar is for a pandemic influenza vaccine. That has contracts, facilities and materiel in place. We saw its implementation with the H1N1 pandemic in 2009. But pandemic influenza does not have the same kind of issues, since it is building on the exisitng work of influenza vaccines, just rapidly creating one for the specific strain in question. So its not "new', its a matter of time, scale, cost and deployment.
