I think the explanation for "oddity" in those numbers (besides the fact that they don't capture causality) is that in usual circumstances vaccines are mostly administered to children, so most deaths are in that group. From a paper that looked at a similar (albeit not exactly the same VAERS year range):
Deaths Reported to the Vaccine Adverse Event Reporting System, United States, 1997–2013
VAERS received 2149 death reports, most (n = 1469 [68.4%]) in children. Median age was 0.5 years (range, 0–100 years). [...] Most common causes of death among 1244 child reports with available death certificates/autopsy reports included sudden infant death syndrome (n = 544 [44%]), asphyxia (n = 74 [6.0%]), [...]
Because SIDS peaks at a time when children are receiving many recommended vaccinations, it would not be unexpected to observe a coincidental close temporal relationship between vaccination and SIDS.
[...] VAERS generally cannot assess if a vaccine caused an adverse event. VAERS does not collect data on the number of individuals vaccinated; therefore, with no denominator data, it is not possible to calculate rates of adverse events. Likewise, VAERS does not collect data on the total number of vaccinated individuals who died; therefore, it is not possible to calculate death rates following vaccination.
Because a large number of vaccines are given to young children (often simultaneously) at scheduled well-child visits, especially during the first year of life, deaths occurring in close temporal association following vaccination are likely to occur by chance alone.
Clearly Covid-19 vaccines have a different demographic profile. In an April 2 Reuters fact-checking article it's been pointed out for instance that
According to the latest data (here), 1,985 U.S. deaths of individuals who died after receiving at least one dose of the COVID-19 vaccines have been reported to VAERS. Of these, 1,016 are listed as receiving doses from Moderna, 946 as Pfizer-BioNTech, 16 as Janssen, and seven as “unknown manufacturer.” In total, 1,579, or nearly early 80%, of these individuals were over the age of 65. [...]
In a WebMD interview here , Dr. Joël Belmin, head of geriatrics and vaccination coordinator at l’hôpital Charles-Foix in Paris, said, “In older people, due to their great frailty, a significant amount of spontaneous mortality is expected. In a retirement home, one in five people die each year. It’s therefore difficult to directly attribute these deaths to the fact that these people were vaccinated.”
So it's apples-to-oranges to compare deaths that predominantly occurred in such an 65+ age group to deaths that mostly occurred in children under one year (as in previous years' VAERS data).
One theory that sometimes explains a higher number of reports for some vaccines is mentioned (with some empirical support) in the 1st paper--the Weber effect, which is generally applicable to new procedures:
We noted that death reports appear to follow the Weber effect, a tendency for new medical products or products perceived to be new to have higher reporting rates for adverse events initially, which then decline despite steadily increasing prescribing rates. For example, the peak in number of death reports during 2001 appears to coincide with an increase in PCV7 use following its licensure and recommendation for use in 2000. RV5 was licensed and recommended in 2006, and the peak in the number of death reports after RV5 occurred in 2008. DTaP-HepB-IPV was first licensed and recommended in 2002 and the first death reports in VAERS were observed in 2003 with the highest number of reports in 2007, which was followed by a decline in subsequent years.
The Weber effect has been correlated with the degree of media coverage as well.
It also has been suggested as such (albeit without naming it) in re Covid-19 vaccines:
experts said it’s not surprising that more death reports would be submitted to VAERS after COVID-19 immunizations compared with flu, given the increased attention on the vaccine.
“The difference in deaths reported following COVID-19 vaccine and flu vaccine administration is likely due to the enhanced reporting of all events occurring after COVID vaccination compared to after influenza vaccination,” Dr. Robert Legare Atmar, an infectious disease specialist at Baylor College of Medicine who also evaluates vaccines, told us.
As there's not much longitudinal data with Covid-19 vaccines, it's probably not yet possible to show a clear trend of over time (with purpose of sense some Weber effect).
A couple of extra points (based on question-comments below):
Although influenza vaccines are administered to the elderly in large numbers in the US, unlike for Covid-19 vaccines, there's no mandatory reporting by healthcare provides of all deaths (to VAERS) after an influenza vaccine; CJR's answer below has more details on that. One could say this is a "forced Weber" effect...
The numbers on both sides of the comparison look plausible insofar as actually being from VAERS as they roughly correspond to figures from other reports/news (which did not attempt such a cross-temporal comparison though). This doesn't make the comparison attempted in the OP's source any more sound though.