In short: no, there's no supported means to disable this.
Terminal automatically applies a minimum contrast when displaying an ANSI (or extended 256-color table) color on the terminal background color, or when displaying the terminal foreground/text color on an ANSI background color.
This is meant to be a convenience, so that one can set a background or foreground color and have them contrast with the ANSI colors without having to adjust all the ANSI colors. This was especially important in earlier versions of Terminal, which lacked Preferences support for customizing the ANSI colors.
Older versions of Terminal applied the minimum-contrast for all color combinations, but that meant that, e.g., ANSI red-on-red text could be readable, but some programs intentionally display text with the same foreground and background color in order to hide it (for displaying game hints or joke punchlines, for example). Because of that, newer versions of Terminal do not apply the minimum-contrast when displaying one ANSI color upon another.
Now, rather than just address the same-color case, Terminal avoids applying the minimum-contrast for all combinations of ANSI colors, because it is assumed that if the user customizes any ANSI colors they will customize them all to contrast with each other as desired. It is also assumed that those users will adjust or take into account the terminal background and foreground colors to fit their desired color scheme.
If the workaround of using colors with different color spaces—suggested in other answer(s)—works, it's likely relying upon a bug in the color-handling code, which is supposed to work with any color, and may stop working in the future when that bug is fixed.
If being able to explicitly disable or adjust the minimum-contrast behavior is important to you, I recommend letting Apple know by filing a report at https://developer.apple.com/bug-reporting/ using any Apple ID.