Can the canon r5 and r6 connect to a phone without that phone disconnecting from the internet

This is most likely an issue on the phone end, not the camera end. Most (all?) phones assume that if WiFi is enabled, that it can use that for internet connectivity, instead of shuffling internet stuff through the cellular network. I have not seen a phone yet that will continue to use e.g. 4G/5G cellular connections for internet access while WiFi is attached to something else. That's not to say that it couldn't be done - just that the software on phones these days just isn't written that way. Maybe an interesting thing to suggest/request to the Android or IOS developers....

Using som old cameras I hade to connect to the cameras built in wifi network to remote control it and download images to my smartphone. This the resulted in my smartphone loosing internet access until I disconnected.

Have this been fixed in more recent models?

According to this article you can have both a wifi connection and internet connection over cellular network in parallel since at least Android 8 and iOS 12.

To clarify: I do not neccecarily mean connect with wifi, I am asking if there is any method to do so, bluthoot or I guess a usb cable would also be considered capable

An USB OTG cable will allow you to connect a card reader to your phone just like you would connect it to a computer and transfer files from the card to your phone with a suitable app, if your phone supports that (many modern smartphones do)

You can also connect to the usb port of the camera directly of course.

A few can. For instance, for Canon cameras that have an external Wireless File Transmitter (WFT) module available, the external WFT module will allow an encrypted connection to an external WiFi network. Maybe it will work with your phone's "hotspot". Though most users seem to say it works better when the AFT module is connected to the same WiFi network as the one to which the phone is connected.

Even those Canon cameras which allow logging onto an external network using an external WFT module do not allow the same using the camera's internal WiFi radio, if the camera has such an internal radio. The internal radio can only host a WiFi network but not connect as a client to another network.

The WFT modules for cameras that do not have a built-in RJ-45 ethernet port also provide one.

Current and past cameras with WFT modules available include the 1-Series bodies (EOS 1D X Mark III, EOS 1D Mark IV, etc.), recent 5-Series bodies (EOS 5D Mark IV, 5D Mark III, etc.), the EOS 7D Mark II, and the just released EOS R5 and R6 mirrorless bodies. Some of the C-series of cinema cameras, such as the C300 Mark II, can also use WFT modules. For the 1-Series/C-series the WFTs plug directly into a proprietary port on the camera. For the 5D/7D series, it's a brick that connects via USB3 which can be used as a dongle or attached to the base of the camera using the 1/4"-16 tripod receiver. For the R5 and R6 it is in the form of battery grip with vertical controls as well as the WiFi functionality and an RJ-45 port for a wired ethernet connection.

Be prepared to pay for the privilege, though. Most of these cameras are not exactly cheap, and the WFT modules themselves run in the high three figures.

On the other end of the connection, there are a few smartphones that have the ability to maintain two connections at the same time. This allows them to join the camera's ad hoc network using one transmitter/antenna while connecting to the internet via WiFi or cellular network (5G, 4G, LTE+, etc.) with another transmitter/antenna. These rare phones aren't exactly cheap, either.