The earliest mention I can find is the SGML standard (1986) which lists intcal in its additional mathematical symbols list under "binary and large operators" the definition being (in full)
<!ENTITY intcal SDATA "[intcal]"--/intercal B: intercal-->
which means the the entity ⊺ could be used to make an unspecified system specific character (with as for all characters in this standard no encoding or pictorial clue what the character should look like). Note however the section heading and the /B in the entity definition comment both imply that the intended use is as a binary operator (which would imply that using it as a funky T for matrix transpose is a mis-use)
Unicode 1.0 added INTERCALATE at U+22BA citing SGML intcal as a prior character
https://www.unicode.org/versions/Unicode1.0.0/ch06.pdf
In MathML we assigned alias ⊺ to this and defined it (as required by XML which does not have SDATA entities) to U+22BA so MathML defines
<!ENTITY intcal "⊺" ><!--INTERCALATE -->
<!ENTITY intercal "⊺" ><!--INTERCALATE -->
so both of names ⊺ and ⊺ work in MathML, and so in HTML since version 5, which incorporated the MathML entities.
Separately the AMS (AMS this time standing for American Mathematical Society, not the AMS in ISOAMSB which stands for ISO Additional Mathematical Symbols set B) produced the TeX AMS Fonts and assigned the character the name \intercal as a binary operator with definition (in its later latex2e form)
\DeclareMathSymbol{\intercal} {\mathbin}{AMSa}{"7C}
The unicode math package keeps the \intercal name defining it, as in MathML, to be U+22BA, again as a binary operator
\UnicodeMathSymbol{"022BA}{\intercal }{\mathbin}{intercal}%
So U+22BA is intercal and looks like ⊺ which is sort of like a squashed dropped sans serif T, not to be confused with T (U+0054) which looks like T, or top (U+22A4) which looks like ⊤.
As the editor of at least some of the specifications mentioned above I can confirm that the name has just been inherited without any actual knowledge of what the character is supposed to be used for.
Wikipedia suggests "intercalate" may be used in
timekeeping
Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.
Chemistry
In chemistry, intercalation is the reversible inclusion or insertion of a molecule (or ion) into materials with layered structures.
University Admin
Intercalation, in the context of university administration, is a period when a student is allowed to officially take time away from studying for an academic degree.
But nowhere do I see any use of a symbol resembling T
\intercalcompletely unapt for the transpose: it's too low.