Are they generally pronounced with the same vowel, or not?
Yes, in most areas, red and read (past tense) are pronounced with the exact same vowel phoneme (which we can write "/ɛ/") and are homophones. The exact phonetic quality of this vowel phoneme /ɛ/ may vary between speakers, or even from time to time with the same speaker. For example, some accents of English use a variant that is phonetically a closer/higher vowel [e], and some accents use a more backed variant, [ɜ] or [ɐ], as a pronunciation of this same vowel phoneme (the vowel found in words like dress).
The category of words that dictionaries show with /ɛ/ has apparently split for some speakers of Singapore English into two categories, one with a higher/closer and one with a lower/opener quality. When you mentioned in a comment that you speak this variety of English, it reminded me that I had heard of this. I cannot remember where I originally read about it, but you can see a presentation covering this phenomenon here: "The NEXT-TEXT split in Singapore English: Comparing self-report and speech production", Rebecca Lurie Starr and Amanda Choo Shimin, (National University of Singapore), presented at New Ways of Analyzing Variation 48 at the University of Oregon, October 2019.
The presentation has a citation indicating that this is apparently not connected to the pronunciation of these words in varieties of English spoken elsewhere:
“there seems to be no straightforward way to predict which vowel some words will have on the basis of BrE or any other external variety of the language. (Deterding 2005:185)”
(page 10)
To me, as a non-speaker of Singapore English, this split is surprising and not something that I would usually think to listen for.