I mean, they're both time traveling phone booths.... Doctor Who obviously predates Bill and Ted's Excellent Adventure, but was also possibly quite obscure for Americans pre-Internet.
Co-writer Chris Matheson claims that neither he, nor co-writer Ed Solomon nor the film's director Stephen Herek (who suggested the phone box) knew anything about Doctor Who.
“Rufus was originally their friend,” reveals Matheson of Bill and Ted’s time-travel mentor, played cooly by comedian George Carlin. “He was a 27-year-old high school junior and he had a dog named rufus. He drove them through time in his 1969 Chevy van. That got changed along the way, basically because of Back to the Future. Director Stephen Herek had the idea of a phone booth and all of us were so clueless and knew nothing about Doctor Who,” laughs Matheson. “It turned out to be pretty good idea. It worked.”
Ed Solomon also addressed the fan-theory that the script was recycled from an earlier Doctor Who script (noting that the Doctor Who film was also produced by the same studio) and made into Bill and Ted.
Q: Is there any truth that original concept for B&T started out originally as a doctor who movie? Can’t remember where I got that idea but the phone booth / time seems to at least factor in the slightest bit. Rufus as a time lord makes me smile
Ed Solomon: No - none at all. The phone booth came later (via the director, Steve Herek, when the studio told us we had to get rid of the van so it didn't seem too much like Back to the Future, which had just come out).
The writers and director claim that they didn't know about the TARDIS. To decide whether that is plausible or not, it is worth considering the differences between the two:
The Doctor's TARDIS is not a public phone box - it is a British Police box. The time machine in Bill & Ted is based on a regular public phone box.
The telephone on a Police box is located on the exterior of the box and cannot make public calls - it is a direct line to the police. The interior of a Police box was "for use by police officers to read and fill in reports, take meal breaks and even temporarily hold detainees until the arrival of transport." (from Wikipedia) There was no phone inside it.
The telephone on the exterior of the TARDIS is not in any way used to pilot it through time. In fact, it wasn't even suggested that the phone had any function at all until way after the first Bill & Ted movie emerged. In contrast, the phone in Bill & Ted's time machine is the means by which they select the time they want to go to.
The TARDIS is "bigger on the inside", whereas the Bill & Ted movies derive a lot of humour from the fact that they squeeze so many people into the tiny phone box.
Although I completely agree that many people would succinctly describe the Doctor's TARDIS as "a phone box" and that this promotes a superficial similarity between the two, there are clearly more differences than similarities which makes the claim of the writers that there was no intended reference plausible.