This is a good question. As detailed below, there is reason to think that mineral oils are unlikely to be compatible between brands. I can't unequivocally show that Shimano oil will damage Tektro brakes. However, if the tradeoff is possible damage to the seals in the Tektro brake, then it would seem better to use Tektro oil.
First, mineral oil is, per Wikipedia:
Mineral oil is any of various colorless, odorless, light mixtures of higher alkanes from a mineral source, particularly a distillate of petroleum...
Higher alkanes are molecules with 9 or more carbon atoms. Per the table in the link, their boiling points range from 150.8 (C9H20) to 286.8 (C16H34) degrees Celsius. Their density varies less, their viscosity varies more.
We don't know the mix of alkanes in Shimano and Tektro oil. Their compositions and resulting physical properties may be close enough for them to be mutually exchangeable. Some posters on bike forums have reported interchanging oils for Shimano and Tektro, e.g. this poster, this poster who said their LBS said the oils are interchangeable, and one poster here who put Shimano oil into Tektro Hy/Rd cable-actuated disc brakes. This article on Juiced Bikes seems to imply that either oil is fine.
Bikerumor interviewed several brake manufacturers, and one of the questions was interchangeability. Some brakes (e.g. SRAM, Hayes, and Hope), use DOT brake fluid (of which there are several types). We can categorically say that DOT fluid should never, ever go into brakes designed for mineral oil. DOT fluid is corrosive to things like paint. If the seals are specifically designed to resist it, this is fine. If they are not, you'll corrode the seals and then your brake will fail.
Shimano said that their brakes could only take Shimano oil:
I can tell you with absolute certainty that Magura fluid will destroy Shimano brakes in a very short amount of time. I’ve seen it happen several times. The rubber seals in the system have to be specifically designed to interact with a specific brake fluid. If you use a different fluid, the seals will interact differently. Specifically, when you put Magura fluid in Shimano brake, the lever feels spongy and the pad contact point changes because the square edge seal at the caliper is breaking free from the piston at a different time.
There are third party companies that make mineral oil brake fluid and
it says right on the bottle that it is compatible with all mineral
brakes. Shimano has never approved a third party brake fluid to be
compatible with our brakes and we never will. How is it possible that
one fluid can work in both Magura and Shimano brakes if Magura fluid
destroys Shimano brakes?
Magura simply said to only use Magura oil in their brakes. However, both manufacturers have a commercial incentive to say that, and Tektro didn't respond to the interview. This leaves open the possibility that Tektro's oil is similar in composition, chemical, and physical properties to Shimano's oil.
Nevertheless, even considering that issue and the forum posts, I don't consider the tradeoff to be worth experimenting. Brakes, after all, stop your bike.